## S2m: Coupled mass-spring systems

#### Example 1

## S2m: Coupled mass-spring systems (ver. 1)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(5\) meters inwards from its natural position, while the outer mass is moved \(\frac{29}{4}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -5 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{29}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{3}{4} \, \cos\left(2 \, \sqrt{5} t\right) - 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-2.91\) meters from its natural position, and the outer mass is located approximately \(-3.23\) meters from its natural position.

#### Example 2

## S2m: Coupled mass-spring systems (ver. 2)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(3\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 3 ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, t\right) + 3 \, \cos\left(t\right)\]

\[x_2= -3 \, \cos\left(2 \, t\right) + 6 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.67\) meters from its natural position, and the outer mass is located approximately \(4.22\) meters from its natural position.

#### Example 3

## S2m: Coupled mass-spring systems (ver. 3)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(10\) meters outwards from its natural position, while the outer mass is moved \(\frac{75}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= 10 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{75}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) + 5 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{5}{4} \, \cos\left(2 \, \sqrt{5} t\right) + 20 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(5.63\) meters from its natural position, and the outer mass is located approximately \(8.48\) meters from its natural position.

#### Example 4

## S2m: Coupled mass-spring systems (ver. 4)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(10\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -10 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{3} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -2 \, \cos\left(2 \, \sqrt{3} t\right) - 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-0.490\) meters from its natural position, and the outer mass is located approximately \(-6.95\) meters from its natural position.

#### Example 5

## S2m: Coupled mass-spring systems (ver. 5)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(14\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 14 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{3} t\right) + 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 2 \, \cos\left(2 \, \sqrt{3} t\right) + 12 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-6.04\) meters from its natural position, and the outer mass is located approximately \(-9.78\) meters from its natural position.

#### Example 6

## S2m: Coupled mass-spring systems (ver. 6)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(5\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 5 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{6} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(0.555\) meters from its natural position, and the outer mass is located approximately \(-3.55\) meters from its natural position.

#### Example 7

## S2m: Coupled mass-spring systems (ver. 7)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(\frac{17}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{17}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{6} t\right) - 5 \, \cos\left(t\right)\]

\[x_2= \frac{3}{2} \, \cos\left(\sqrt{6} t\right) - 10 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-4.27\) meters from its natural position, and the outer mass is located approximately \(-1.41\) meters from its natural position.

#### Example 8

## S2m: Coupled mass-spring systems (ver. 8)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{20}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{20}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{21} t\right) + 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{2}{3} \, \cos\left(\sqrt{21} t\right) + 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(1.96\) meters from its natural position, and the outer mass is located approximately \(-6.35\) meters from its natural position.

#### Example 9

## S2m: Coupled mass-spring systems (ver. 9)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(9\) meters outwards from its natural position, while the outer mass is moved \(8\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(4\) seconds.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 9 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 8 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{6} t\right) + 5 \, \cos\left(t\right)\]

\[x_2= -2 \, \cos\left(\sqrt{6} t\right) + 10 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-6.99\) meters from its natural position, and the outer mass is located approximately \(-4.67\) meters from its natural position.

#### Example 10

## S2m: Coupled mass-spring systems (ver. 10)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{33}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(3\) seconds.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{33}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.540\) meters from its natural position, and the outer mass is located approximately \(-7.77\) meters from its natural position.

#### Example 11

## S2m: Coupled mass-spring systems (ver. 11)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(7\) meters inwards from its natural position, while the outer mass is moved \(2\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(2\) seconds.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -7 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= 2 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{5} t\right) - 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 6 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(6.39\) meters from its natural position, and the outer mass is located approximately \(-1.53\) meters from its natural position.

#### Example 12

## S2m: Coupled mass-spring systems (ver. 12)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(\frac{34}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{34}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{2}{3} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-7.67\) meters from its natural position, and the outer mass is located approximately \(-10.8\) meters from its natural position.

#### Example 13

## S2m: Coupled mass-spring systems (ver. 13)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(\frac{15}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{15}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(\sqrt{10} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(0.334\) meters from its natural position, and the outer mass is located approximately \(-3.83\) meters from its natural position.

#### Example 14

## S2m: Coupled mass-spring systems (ver. 14)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(9\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -9 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{10} t\right) - 4 \, \cos\left(t\right)\]

\[x_2= -\cos\left(\sqrt{10} t\right) - 8 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.0322\) meters from its natural position, and the outer mass is located approximately \(8.92\) meters from its natural position.

#### Example 15

## S2m: Coupled mass-spring systems (ver. 15)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{25}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{25}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -3 \, \cos\left(2 \, \sqrt{5} t\right) - \frac{16}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.540\) meters from its natural position, and the outer mass is located approximately \(-4.46\) meters from its natural position.

#### Example 16

## S2m: Coupled mass-spring systems (ver. 16)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{33}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{33}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-2.05\) meters from its natural position, and the outer mass is located approximately \(-13.1\) meters from its natural position.

#### Example 17

## S2m: Coupled mass-spring systems (ver. 17)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(9\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -9 ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, t\right) - 2 \, \cos\left(t\right)\]

\[x_2= -5 \, \cos\left(2 \, t\right) - 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(6.78\) meters from its natural position, and the outer mass is located approximately \(-0.841\) meters from its natural position.

#### Example 18

## S2m: Coupled mass-spring systems (ver. 18)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(5\) meters outwards from its natural position, while the outer mass is moved \(\frac{13}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 5 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{13}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(\sqrt{10} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= -\frac{3}{4} \, \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-4.97\) meters from its natural position, and the outer mass is located approximately \(-3.21\) meters from its natural position.

#### Example 19

## S2m: Coupled mass-spring systems (ver. 19)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{47}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{47}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{15} t\right) - 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{10}{3} \, \cos\left(\sqrt{15} t\right) - \frac{9}{2} \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-7.31\) meters from its natural position, and the outer mass is located approximately \(-0.390\) meters from its natural position.

#### Example 20

## S2m: Coupled mass-spring systems (ver. 20)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(\frac{10}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{10}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{15} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{8}{3} \, \cos\left(\sqrt{15} t\right) + 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-0.667\) meters from its natural position, and the outer mass is located approximately \(7.47\) meters from its natural position.

#### Example 21

## S2m: Coupled mass-spring systems (ver. 21)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{43}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

Give a system of IVPs that models this scenario, then solve the system. Use your solution to find the position of both masses after \(5\) seconds.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{43}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{9}{2} \, \cos\left(2 \, \sqrt{5} t\right) - \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.35\) meters from its natural position, and the outer mass is located approximately \(6.12\) meters from its natural position.

#### Example 22

## S2m: Coupled mass-spring systems (ver. 22)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{25}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{25}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{2} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(4.64\) meters from its natural position, and the outer mass is located approximately \(8.16\) meters from its natural position.

#### Example 23

## S2m: Coupled mass-spring systems (ver. 23)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{25}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{25}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{2} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-2.95\) meters from its natural position, and the outer mass is located approximately \(-5.24\) meters from its natural position.

#### Example 24

## S2m: Coupled mass-spring systems (ver. 24)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(\frac{31}{6}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{31}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{21} t\right) + 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{5}{6} \, \cos\left(\sqrt{21} t\right) + 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(5.96\) meters from its natural position, and the outer mass is located approximately \(4.14\) meters from its natural position.

#### Example 25

## S2m: Coupled mass-spring systems (ver. 25)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(4\) meters inwards from its natural position, while the outer mass is moved \(\frac{17}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -4 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{17}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{21} t\right) - 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{3} \, \cos\left(\sqrt{21} t\right) - 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-3.36\) meters from its natural position, and the outer mass is located approximately \(-4.57\) meters from its natural position.

#### Example 26

## S2m: Coupled mass-spring systems (ver. 26)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(5\) meters inwards from its natural position, while the outer mass is moved \(4\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= -5 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -4 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, t\right) - 3 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(2 \, t\right) - 6 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(0.827\) meters from its natural position, and the outer mass is located approximately \(-3.38\) meters from its natural position.

#### Example 27

## S2m: Coupled mass-spring systems (ver. 27)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(\frac{35}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{35}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{21} t\right) - 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{3} \, \cos\left(\sqrt{21} t\right) - 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-4.98\) meters from its natural position, and the outer mass is located approximately \(-9.43\) meters from its natural position.

#### Example 28

## S2m: Coupled mass-spring systems (ver. 28)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(\frac{7}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{7}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -5 \, \cos\left(\sqrt{6} t\right) - 3 \, \cos\left(t\right)\]

\[x_2= \frac{5}{2} \, \cos\left(\sqrt{6} t\right) - 6 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(0.321\) meters from its natural position, and the outer mass is located approximately \(2.96\) meters from its natural position.

#### Example 29

## S2m: Coupled mass-spring systems (ver. 29)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(3\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 3 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{10} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= -\cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(3.16\) meters from its natural position, and the outer mass is located approximately \(-2.66\) meters from its natural position.

#### Example 30

## S2m: Coupled mass-spring systems (ver. 30)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(\frac{31}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{31}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{1}{2} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(4.75\) meters from its natural position, and the outer mass is located approximately \(11.1\) meters from its natural position.

#### Example 31

## S2m: Coupled mass-spring systems (ver. 31)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{47}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{47}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{15} t\right) - 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{10}{3} \, \cos\left(\sqrt{15} t\right) - \frac{9}{2} \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-7.31\) meters from its natural position, and the outer mass is located approximately \(-0.390\) meters from its natural position.

#### Example 32

## S2m: Coupled mass-spring systems (ver. 32)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(6\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 6 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{6} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-3.92\) meters from its natural position, and the outer mass is located approximately \(-2.99\) meters from its natural position.

#### Example 33

## S2m: Coupled mass-spring systems (ver. 33)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(7\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, t\right) - 5 \, \cos\left(t\right)\]

\[x_2= 3 \, \cos\left(2 \, t\right) - 10 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(3.70\) meters from its natural position, and the outer mass is located approximately \(6.10\) meters from its natural position.

#### Example 34

## S2m: Coupled mass-spring systems (ver. 34)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(7\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{6} t\right) - 4 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{6} t\right) - 8 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(1.29\) meters from its natural position, and the outer mass is located approximately \(3.51\) meters from its natural position.

#### Example 35

## S2m: Coupled mass-spring systems (ver. 35)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{19}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= \frac{19}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= \frac{3}{2} \, \cos\left(\sqrt{6} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.71\) meters from its natural position, and the outer mass is located approximately \(3.69\) meters from its natural position.

#### Example 36

## S2m: Coupled mass-spring systems (ver. 36)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(0\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 0 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, t\right) + 2 \, \cos\left(t\right)\]

\[x_2= -4 \, \cos\left(2 \, t\right) + 4 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-1.89\) meters from its natural position, and the outer mass is located approximately \(-2.03\) meters from its natural position.

#### Example 37

## S2m: Coupled mass-spring systems (ver. 37)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{10} t\right) + 3 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{10} t\right) + 6 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-5.95\) meters from its natural position, and the outer mass is located approximately \(-2.93\) meters from its natural position.

#### Example 38

## S2m: Coupled mass-spring systems (ver. 38)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(4\) meters outwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 4 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{3} t\right) + 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\cos\left(2 \, \sqrt{3} t\right) + 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.36\) meters from its natural position, and the outer mass is located approximately \(-5.81\) meters from its natural position.

#### Example 39

## S2m: Coupled mass-spring systems (ver. 39)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(\frac{61}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{61}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{15}{2} \, \cos\left(2 \, \sqrt{5} t\right) - \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-2.54\) meters from its natural position, and the outer mass is located approximately \(9.18\) meters from its natural position.

#### Example 40

## S2m: Coupled mass-spring systems (ver. 40)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(15\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 15 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{5} t\right) + 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\cos\left(2 \, \sqrt{5} t\right) + 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-7.34\) meters from its natural position, and the outer mass is located approximately \(-14.3\) meters from its natural position.

#### Example 41

## S2m: Coupled mass-spring systems (ver. 41)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\cos\left(\sqrt{6} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-4.48\) meters from its natural position, and the outer mass is located approximately \(-4.30\) meters from its natural position.

#### Example 42

## S2m: Coupled mass-spring systems (ver. 42)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(10\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= 10 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{5} t\right) + 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 6 \, \cos\left(2 \, \sqrt{5} t\right) + 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(1.56\) meters from its natural position, and the outer mass is located approximately \(-8.48\) meters from its natural position.

#### Example 43

## S2m: Coupled mass-spring systems (ver. 43)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{11}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{11}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(\sqrt{6} t\right) - 2 \, \cos\left(t\right)\]

\[x_2= -\frac{3}{2} \, \cos\left(\sqrt{6} t\right) - 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(3.43\) meters from its natural position, and the outer mass is located approximately \(3.23\) meters from its natural position.

#### Example 44

## S2m: Coupled mass-spring systems (ver. 44)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(5\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 5 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{10} t\right) + 2 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(2.01\) meters from its natural position, and the outer mass is located approximately \(-4.96\) meters from its natural position.

#### Example 45

## S2m: Coupled mass-spring systems (ver. 45)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{22}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{22}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{15} t\right) - 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{4}{3} \, \cos\left(\sqrt{15} t\right) - 6 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(2.98\) meters from its natural position, and the outer mass is located approximately \(1.94\) meters from its natural position.

#### Example 46

## S2m: Coupled mass-spring systems (ver. 46)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(\frac{25}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{25}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{21} t\right) + 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{2}{3} \, \cos\left(\sqrt{21} t\right) + 9 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(0.160\) meters from its natural position, and the outer mass is located approximately \(-4.33\) meters from its natural position.

#### Example 47

## S2m: Coupled mass-spring systems (ver. 47)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(10\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -10 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{5} t\right) - 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -6 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-0.702\) meters from its natural position, and the outer mass is located approximately \(9.11\) meters from its natural position.

#### Example 48

## S2m: Coupled mass-spring systems (ver. 48)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(\frac{61}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{61}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{15}{2} \, \cos\left(2 \, \sqrt{5} t\right) - \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(2.37\) meters from its natural position, and the outer mass is located approximately \(-6.19\) meters from its natural position.

#### Example 49

## S2m: Coupled mass-spring systems (ver. 49)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(12\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 12 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, t\right) + 5 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(2 \, t\right) + 10 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-6.87\) meters from its natural position, and the outer mass is located approximately \(-7.98\) meters from its natural position.

#### Example 50

## S2m: Coupled mass-spring systems (ver. 50)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(\frac{45}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{45}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{3} t\right) - 5 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{5}{2} \, \cos\left(2 \, \sqrt{3} t\right) - 20 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-2.61\) meters from its natural position, and the outer mass is located approximately \(-16.7\) meters from its natural position.

#### Example 51

## S2m: Coupled mass-spring systems (ver. 51)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(9\) meters inwards from its natural position, while the outer mass is moved \(\frac{59}{4}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -9 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{59}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= -5 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{5}{4} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-5.16\) meters from its natural position, and the outer mass is located approximately \(-6.62\) meters from its natural position.

#### Example 52

## S2m: Coupled mass-spring systems (ver. 52)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(7\) meters inwards from its natural position, while the outer mass is moved \(2\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= -7 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -2 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, t\right) - 3 \, \cos\left(t\right)\]

\[x_2= 4 \, \cos\left(2 \, t\right) - 6 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-0.871\) meters from its natural position, and the outer mass is located approximately \(9.78\) meters from its natural position.

#### Example 53

## S2m: Coupled mass-spring systems (ver. 53)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(14\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -14 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{3} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 2 \, \cos\left(2 \, \sqrt{3} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(0.597\) meters from its natural position, and the outer mass is located approximately \(16.8\) meters from its natural position.

#### Example 54

## S2m: Coupled mass-spring systems (ver. 54)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(\frac{7}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= \frac{7}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(2 \, \sqrt{5} t\right) + 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -3 \, \cos\left(2 \, \sqrt{5} t\right) + \frac{16}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(3.18\) meters from its natural position, and the outer mass is located approximately \(0.501\) meters from its natural position.

#### Example 55

## S2m: Coupled mass-spring systems (ver. 55)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(9\) meters outwards from its natural position, while the outer mass is moved \(\frac{37}{6}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 10 x_1+ 6 x_2\hspace{2em}x_1(0)= 9 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{37}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{3} t\right) + 5 \, \cos\left(t\right)\]

\[x_2= -\frac{4}{3} \, \cos\left(2 \, \sqrt{3} t\right) + \frac{15}{2} \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-2.16\) meters from its natural position, and the outer mass is located approximately \(-5.27\) meters from its natural position.

#### Example 56

## S2m: Coupled mass-spring systems (ver. 56)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(1\) meters outwards from its natural position, while the outer mass is moved \(\frac{31}{6}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 10 x_1+ 6 x_2\hspace{2em}x_1(0)= 1 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{31}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, \sqrt{3} t\right) + 3 \, \cos\left(t\right)\]

\[x_2= \frac{2}{3} \, \cos\left(2 \, \sqrt{3} t\right) + \frac{9}{2} \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-1.84\) meters from its natural position, and the outer mass is located approximately \(-4.83\) meters from its natural position.

#### Example 57

## S2m: Coupled mass-spring systems (ver. 57)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(\frac{13}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= \frac{13}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(\sqrt{6} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\frac{3}{2} \, \cos\left(\sqrt{6} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-5.41\) meters from its natural position, and the outer mass is located approximately \(-3.83\) meters from its natural position.

#### Example 58

## S2m: Coupled mass-spring systems (ver. 58)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{17}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 10 x_1+ 6 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{17}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, \sqrt{3} t\right) - 5 \, \cos\left(t\right)\]

\[x_2= -\cos\left(2 \, \sqrt{3} t\right) - \frac{15}{2} \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(4.48\) meters from its natural position, and the outer mass is located approximately \(2.32\) meters from its natural position.

#### Example 59

## S2m: Coupled mass-spring systems (ver. 59)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(\frac{15}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{15}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(\sqrt{10} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-0.855\) meters from its natural position, and the outer mass is located approximately \(2.77\) meters from its natural position.

#### Example 60

## S2m: Coupled mass-spring systems (ver. 60)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(7\) meters inwards from its natural position, while the outer mass is moved \(\frac{19}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 10 x_1+ 6 x_2\hspace{2em}x_1(0)= -7 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{19}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{3} t\right) - 3 \, \cos\left(t\right)\]

\[x_2= \frac{4}{3} \, \cos\left(2 \, \sqrt{3} t\right) - \frac{9}{2} \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.02\) meters from its natural position, and the outer mass is located approximately \(-1.22\) meters from its natural position.

#### Example 61

## S2m: Coupled mass-spring systems (ver. 61)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(\frac{5}{6}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= \frac{5}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, \sqrt{5} t\right) + 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{9}{2} \, \cos\left(2 \, \sqrt{5} t\right) + \frac{16}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(3.84\) meters from its natural position, and the outer mass is located approximately \(-0.490\) meters from its natural position.

#### Example 62

## S2m: Coupled mass-spring systems (ver. 62)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(5\) meters inwards from its natural position, while the outer mass is moved \(1\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= -5 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -1 ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(\sqrt{15} t\right) - 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= 2 \, \cos\left(\sqrt{15} t\right) - 3 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(1.58\) meters from its natural position, and the outer mass is located approximately \(3.07\) meters from its natural position.

#### Example 63

## S2m: Coupled mass-spring systems (ver. 63)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(\frac{49}{6}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{49}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{21} t\right) + 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{5}{6} \, \cos\left(\sqrt{21} t\right) + 9 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(0.539\) meters from its natural position, and the outer mass is located approximately \(-4.39\) meters from its natural position.

#### Example 64

## S2m: Coupled mass-spring systems (ver. 64)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{5} t\right) + 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\cos\left(2 \, \sqrt{5} t\right) + 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-5.17\) meters from its natural position, and the outer mass is located approximately \(-4.84\) meters from its natural position.

#### Example 65

## S2m: Coupled mass-spring systems (ver. 65)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(3\) meters outwards from its natural position, while the outer mass is moved \(\frac{37}{4}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= 3 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{37}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{5}{4} \, \cos\left(2 \, \sqrt{5} t\right) - 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(1.27\) meters from its natural position, and the outer mass is located approximately \(-7.11\) meters from its natural position.

#### Example 66

## S2m: Coupled mass-spring systems (ver. 66)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{67}{6}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= \frac{67}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, \sqrt{5} t\right) + 5 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{9}{2} \, \cos\left(2 \, \sqrt{5} t\right) + \frac{20}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(0.345\) meters from its natural position, and the outer mass is located approximately \(6.07\) meters from its natural position.

#### Example 67

## S2m: Coupled mass-spring systems (ver. 67)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{29}{4}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{29}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{10} t\right) - 3 \, \cos\left(t\right)\]

\[x_2= -\frac{5}{4} \, \cos\left(\sqrt{10} t\right) - 6 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-2.02\) meters from its natural position, and the outer mass is located approximately \(7.19\) meters from its natural position.

#### Example 68

## S2m: Coupled mass-spring systems (ver. 68)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(7\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{6} t\right) - 4 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{6} t\right) - 8 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(4.48\) meters from its natural position, and the outer mass is located approximately \(4.30\) meters from its natural position.

#### Example 69

## S2m: Coupled mass-spring systems (ver. 69)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(11\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -11 ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, t\right) - 3 \, \cos\left(t\right)\]

\[x_2= -5 \, \cos\left(2 \, t\right) - 6 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-2.02\) meters from its natural position, and the outer mass is located approximately \(5.76\) meters from its natural position.

#### Example 70

## S2m: Coupled mass-spring systems (ver. 70)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(\frac{13}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{13}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{10} t\right) - 3 \, \cos\left(t\right)\]

\[x_2= -\frac{1}{2} \, \cos\left(\sqrt{10} t\right) - 6 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(0.974\) meters from its natural position, and the outer mass is located approximately \(6.44\) meters from its natural position.

#### Example 71

## S2m: Coupled mass-spring systems (ver. 71)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(\frac{38}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{38}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{5} t\right) - 5 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -6 \, \cos\left(2 \, \sqrt{5} t\right) - \frac{20}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-1.70\) meters from its natural position, and the outer mass is located approximately \(-8.76\) meters from its natural position.

#### Example 72

## S2m: Coupled mass-spring systems (ver. 72)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(7\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= 7 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{10} t\right) + 3 \, \cos\left(t\right)\]

\[x_2= \cos\left(\sqrt{10} t\right) + 6 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(1.02\) meters from its natural position, and the outer mass is located approximately \(-6.94\) meters from its natural position.

#### Example 73

## S2m: Coupled mass-spring systems (ver. 73)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(2\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -2 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(\sqrt{6} t\right) - 2 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(\sqrt{6} t\right) - 4 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(0.0436\) meters from its natural position, and the outer mass is located approximately \(4.93\) meters from its natural position.

#### Example 74

## S2m: Coupled mass-spring systems (ver. 74)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(10\) meters inwards from its natural position, while the outer mass is moved \(5\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= -10 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -5 ,x_2'(0)=0\]

This system solves to:

\[x_1= -5 \, \cos\left(2 \, t\right) - 5 \, \cos\left(t\right)\]

\[x_2= 5 \, \cos\left(2 \, t\right) - 10 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(4.00\) meters from its natural position, and the outer mass is located approximately \(5.81\) meters from its natural position.

#### Example 75

## S2m: Coupled mass-spring systems (ver. 75)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(7\) meters inwards from its natural position, while the outer mass is moved \(\frac{3}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -7 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -\frac{3}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -5 \, \cos\left(\sqrt{6} t\right) - 2 \, \cos\left(t\right)\]

\[x_2= \frac{5}{2} \, \cos\left(\sqrt{6} t\right) - 4 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(5.96\) meters from its natural position, and the outer mass is located approximately \(0.286\) meters from its natural position.

#### Example 76

## S2m: Coupled mass-spring systems (ver. 76)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(4\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 4 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{6} t\right) + 3 \, \cos\left(t\right)\]

\[x_2= -2 \, \cos\left(\sqrt{6} t\right) + 6 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-5.69\) meters from its natural position, and the outer mass is located approximately \(-2.06\) meters from its natural position.

#### Example 77

## S2m: Coupled mass-spring systems (ver. 77)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(\frac{61}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{61}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{15} t\right) - 5 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{8}{3} \, \cos\left(\sqrt{15} t\right) - \frac{15}{2} \, \cos\left(\sqrt{2} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-0.0481\) meters from its natural position, and the outer mass is located approximately \(-7.61\) meters from its natural position.

#### Example 78

## S2m: Coupled mass-spring systems (ver. 78)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(9\) meters outwards from its natural position, while the outer mass is moved \(\frac{27}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 9 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{27}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\frac{5}{4} \, \cos\left(\sqrt{10} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-8.95\) meters from its natural position, and the outer mass is located approximately \(-6.67\) meters from its natural position.

#### Example 79

## S2m: Coupled mass-spring systems (ver. 79)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{37}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{37}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(\sqrt{21} t\right) + 4 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= \frac{1}{3} \, \cos\left(\sqrt{21} t\right) + 12 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-2.57\) meters from its natural position, and the outer mass is located approximately \(-5.31\) meters from its natural position.

#### Example 80

## S2m: Coupled mass-spring systems (ver. 80)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(\frac{17}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= \frac{17}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, \sqrt{5} t\right) + 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{9}{2} \, \cos\left(2 \, \sqrt{5} t\right) + 4 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(0.679\) meters from its natural position, and the outer mass is located approximately \(5.77\) meters from its natural position.

#### Example 81

## S2m: Coupled mass-spring systems (ver. 81)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(0\) meters outwards from its natural position, while the outer mass is moved \(12\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 0 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -12 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, t\right) - 4 \, \cos\left(t\right)\]

\[x_2= -4 \, \cos\left(2 \, t\right) - 8 \, \cos\left(t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(7.80\) meters from its natural position, and the outer mass is located approximately \(4.08\) meters from its natural position.

#### Example 82

## S2m: Coupled mass-spring systems (ver. 82)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(14\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(\frac{28}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 20 x_1+ 6 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{28}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{21} t\right) - 3 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{1}{3} \, \cos\left(\sqrt{21} t\right) - 9 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-0.694\) meters from its natural position, and the outer mass is located approximately \(-7.58\) meters from its natural position.

#### Example 83

## S2m: Coupled mass-spring systems (ver. 83)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters inwards from its natural position, while the outer mass is moved \(\frac{17}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 10 x_1+ 6 x_2\hspace{2em}x_1(0)= -2 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{17}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, \sqrt{3} t\right) - 5 \, \cos\left(t\right)\]

\[x_2= -\cos\left(2 \, \sqrt{3} t\right) - \frac{15}{2} \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(4.48\) meters from its natural position, and the outer mass is located approximately \(2.32\) meters from its natural position.

#### Example 84

## S2m: Coupled mass-spring systems (ver. 84)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(10\) N/m, the outer spring has constant \(12\) N/m. The inner mass is moved \(5\) meters outwards from its natural position, while the outer mass is moved \(\frac{11}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 12 x_2\hspace{2em}x_1(0)= 5 ,x_1'(0)=0\]

\[1 x_2''= 12 x_1- 12 x_2\hspace{2em}x_2(0)= -\frac{11}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, \sqrt{5} t\right) + 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{9}{2} \, \cos\left(2 \, \sqrt{5} t\right) + \frac{8}{3} \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(3.32\) meters from its natural position, and the outer mass is located approximately \(-0.447\) meters from its natural position.

#### Example 85

## S2m: Coupled mass-spring systems (ver. 85)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(7\) meters inwards from its natural position, while the outer mass is moved \(\frac{27}{4}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -7 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{27}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= -5 \, \cos\left(2 \, \sqrt{5} t\right) - 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{5}{4} \, \cos\left(2 \, \sqrt{5} t\right) - 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(6.33\) meters from its natural position, and the outer mass is located approximately \(6.48\) meters from its natural position.

#### Example 86

## S2m: Coupled mass-spring systems (ver. 86)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(7\) meters outwards from its natural position, while the outer mass is moved \(\frac{11}{2}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 7 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{11}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{3} t\right) + 2 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{5}{2} \, \cos\left(2 \, \sqrt{3} t\right) + 8 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(2.98\) meters from its natural position, and the outer mass is located approximately \(5.70\) meters from its natural position.

#### Example 87

## S2m: Coupled mass-spring systems (ver. 87)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(\frac{19}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{19}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{10} t\right) + 3 \, \cos\left(t\right)\]

\[x_2= -\frac{5}{4} \, \cos\left(\sqrt{10} t\right) + 6 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(3.02\) meters from its natural position, and the outer mass is located approximately \(-5.17\) meters from its natural position.

#### Example 88

## S2m: Coupled mass-spring systems (ver. 88)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(3\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(1\) meters inwards from its natural position, while the outer mass is moved \(12\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 5 x_1+ 2 x_2\hspace{2em}x_1(0)= -1 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -12 ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(\sqrt{6} t\right) - 5 \, \cos\left(t\right)\]

\[x_2= -2 \, \cos\left(\sqrt{6} t\right) - 10 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(2.82\) meters from its natural position, and the outer mass is located approximately \(3.79\) meters from its natural position.

#### Example 89

## S2m: Coupled mass-spring systems (ver. 89)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(10\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 10 ,x_2'(0)=0\]

This system solves to:

\[x_1= -2 \, \cos\left(2 \, t\right) + 4 \, \cos\left(t\right)\]

\[x_2= 2 \, \cos\left(2 \, t\right) + 8 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-2.32\) meters from its natural position, and the outer mass is located approximately \(-5.52\) meters from its natural position.

#### Example 90

## S2m: Coupled mass-spring systems (ver. 90)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(5\) meters outwards from its natural position, while the outer mass is moved \(1\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 5 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= 1 ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(\sqrt{15} t\right) + 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -2 \, \cos\left(\sqrt{15} t\right) + 3 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-1.31\) meters from its natural position, and the outer mass is located approximately \(4.38\) meters from its natural position.

#### Example 91

## S2m: Coupled mass-spring systems (ver. 91)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(9\) meters inwards from its natural position, while the outer mass is moved \(18\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= -9 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -18 ,x_2'(0)=0\]

This system solves to:

\[x_1= -4 \, \cos\left(2 \, \sqrt{3} t\right) - 5 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= 2 \, \cos\left(2 \, \sqrt{3} t\right) - 20 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(-5.10\) meters from its natural position, and the outer mass is located approximately \(-15.4\) meters from its natural position.

#### Example 92

## S2m: Coupled mass-spring systems (ver. 92)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(8\) meters outwards from its natural position, while the outer mass is moved \(\frac{19}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 8 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{19}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{10} t\right) + 3 \, \cos\left(t\right)\]

\[x_2= -\frac{5}{4} \, \cos\left(\sqrt{10} t\right) + 6 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(3.02\) meters from its natural position, and the outer mass is located approximately \(-5.17\) meters from its natural position.

#### Example 93

## S2m: Coupled mass-spring systems (ver. 93)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters outwards from its natural position, while the outer mass is moved \(3\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= 6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= 3 ,x_2'(0)=0\]

This system solves to:

\[x_1= 3 \, \cos\left(2 \, t\right) + 3 \, \cos\left(t\right)\]

\[x_2= -3 \, \cos\left(2 \, t\right) + 6 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(-3.21\) meters from its natural position, and the outer mass is located approximately \(-0.536\) meters from its natural position.

#### Example 94

## S2m: Coupled mass-spring systems (ver. 94)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(8\) meters inwards from its natural position, while the outer mass is moved \(\frac{17}{6}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 10 x_1+ 6 x_2\hspace{2em}x_1(0)= -8 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{17}{6} ,x_2'(0)=0\]

This system solves to:

\[x_1= -5 \, \cos\left(2 \, \sqrt{3} t\right) - 3 \, \cos\left(t\right)\]

\[x_2= \frac{5}{3} \, \cos\left(2 \, \sqrt{3} t\right) - \frac{9}{2} \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-1.06\) meters from its natural position, and the outer mass is located approximately \(-1.21\) meters from its natural position.

#### Example 95

## S2m: Coupled mass-spring systems (ver. 95)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(9\) meters outwards from its natural position, while the outer mass is moved \(\frac{27}{4}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 9 x_1+ 4 x_2\hspace{2em}x_1(0)= 9 ,x_1'(0)=0\]

\[2 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= \frac{27}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(\sqrt{10} t\right) + 4 \, \cos\left(t\right)\]

\[x_2= -\frac{5}{4} \, \cos\left(\sqrt{10} t\right) + 8 \, \cos\left(t\right)\]

Thus after \(4\) seconds, the inner mass is located approximately \(2.37\) meters from its natural position, and the outer mass is located approximately \(-6.47\) meters from its natural position.

#### Example 96

## S2m: Coupled mass-spring systems (ver. 96)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(15\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(7\) meters inwards from its natural position, while the outer mass is moved \(\frac{61}{4}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 19 x_1+ 4 x_2\hspace{2em}x_1(0)= -7 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{61}{4} ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, \sqrt{5} t\right) - 4 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= \frac{3}{4} \, \cos\left(2 \, \sqrt{5} t\right) - 16 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(-3.84\) meters from its natural position, and the outer mass is located approximately \(-6.95\) meters from its natural position.

#### Example 97

## S2m: Coupled mass-spring systems (ver. 97)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(18\) N/m, the outer spring has constant \(4\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{29}{2}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 22 x_1+ 4 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[1 x_2''= 4 x_1- 4 x_2\hspace{2em}x_2(0)= -\frac{29}{2} ,x_2'(0)=0\]

This system solves to:

\[x_1= 5 \, \cos\left(2 \, \sqrt{3} t\right) - 3 \, \cos\left(\sqrt{3} t\right)\]

\[x_2= -\frac{5}{2} \, \cos\left(2 \, \sqrt{3} t\right) - 12 \, \cos\left(\sqrt{3} t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(2.37\) meters from its natural position, and the outer mass is located approximately \(8.56\) meters from its natural position.

#### Example 98

## S2m: Coupled mass-spring systems (ver. 98)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(2\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(2\) meters outwards from its natural position, while the outer mass is moved \(\frac{13}{3}\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 10 x_1+ 6 x_2\hspace{2em}x_1(0)= 2 ,x_1'(0)=0\]

\[2 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= -\frac{13}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 4 \, \cos\left(2 \, \sqrt{3} t\right) - 2 \, \cos\left(t\right)\]

\[x_2= -\frac{4}{3} \, \cos\left(2 \, \sqrt{3} t\right) - 3 \, \cos\left(t\right)\]

Thus after \(5\) seconds, the inner mass is located approximately \(-0.401\) meters from its natural position, and the outer mass is located approximately \(-0.906\) meters from its natural position.

#### Example 99

## S2m: Coupled mass-spring systems (ver. 99)

Consider a coupled mass-spring system where the inner mass is \(2\) kg, the outer mass is \(1\) kg, the inner spring has constant \(4\) N/m, the outer spring has constant \(2\) N/m. The inner mass is moved \(6\) meters inwards from its natural position, while the outer mass is moved \(3\) meters inwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[2 x_1''=- 6 x_1+ 2 x_2\hspace{2em}x_1(0)= -6 ,x_1'(0)=0\]

\[1 x_2''= 2 x_1- 2 x_2\hspace{2em}x_2(0)= -3 ,x_2'(0)=0\]

This system solves to:

\[x_1= -3 \, \cos\left(2 \, t\right) - 3 \, \cos\left(t\right)\]

\[x_2= 3 \, \cos\left(2 \, t\right) - 6 \, \cos\left(t\right)\]

Thus after \(2\) seconds, the inner mass is located approximately \(3.21\) meters from its natural position, and the outer mass is located approximately \(0.536\) meters from its natural position.

#### Example 100

## S2m: Coupled mass-spring systems (ver. 100)

Consider a coupled mass-spring system where the inner mass is \(1\) kg, the outer mass is \(1\) kg, the inner spring has constant \(5\) N/m, the outer spring has constant \(6\) N/m. The inner mass is moved \(4\) meters outwards from its natural position, while the outer mass is moved \(\frac{5}{3}\) meters outwards from its natural position. Both masses are then simultaneously released from rest.

#### Answer.

The IVP system is given by:

\[1 x_1''=- 11 x_1+ 6 x_2\hspace{2em}x_1(0)= 4 ,x_1'(0)=0\]

\[1 x_2''= 6 x_1- 6 x_2\hspace{2em}x_2(0)= \frac{5}{3} ,x_2'(0)=0\]

This system solves to:

\[x_1= 2 \, \cos\left(\sqrt{15} t\right) + 2 \, \cos\left(\sqrt{2} t\right)\]

\[x_2= -\frac{4}{3} \, \cos\left(\sqrt{15} t\right) + 3 \, \cos\left(\sqrt{2} t\right)\]

Thus after \(3\) seconds, the inner mass is located approximately \(0.263\) meters from its natural position, and the outer mass is located approximately \(-2.14\) meters from its natural position.