C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP.

Example 1

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 1)

A mass of \(25\) kg is attached to a certain spring such that \(48\) Newtons of force is required to compress the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.2\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=-3, x'(0)=0 \]

The position after \(9.2\) seconds is approximately \(1.42\) meters inward.

Example 2

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 2)

A mass of \(25\) kg is attached to a certain spring such that \(18\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.5\) seconds.

Answer.

The IVP is given by

\[ 25x''+9x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(3.5\) seconds is approximately \(1.01\) meters outward.

Example 3

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 3)

A mass of \(9\) kg is attached to a certain spring such that \(125\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.9\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(7.9\) seconds is approximately \(4.12\) meters inward.

Example 4

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 4)

A mass of \(16\) kg is attached to a certain spring such that \(18\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(8.8\) seconds.

Answer.

The IVP is given by

\[ 16x''+9x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(8.8\) seconds is approximately \(1.90\) meters outward.

Example 5

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 5)

A mass of \(4\) kg is attached to a certain spring such that \(32\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(5.5\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(5.5\) seconds is approximately \(0.00885\) meters inward.

Example 6

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 6)

A mass of \(25\) kg is attached to a certain spring such that \(27\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(4.6\) seconds.

Answer.

The IVP is given by

\[ 25x''+9x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(4.6\) seconds is approximately \(2.78\) meters inward.

Example 7

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 7)

A mass of \(25\) kg is attached to a certain spring such that \(80\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(5.9\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(5.9\) seconds is approximately \(0.0391\) meters outward.

Example 8

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 8)

A mass of \(25\) kg is attached to a certain spring such that \(32\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.4\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(9.4\) seconds is approximately \(0.656\) meters outward.

Example 9

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 9)

A mass of \(16\) kg is attached to a certain spring such that \(36\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.9\) seconds.

Answer.

The IVP is given by

\[ 16x''+9x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(2.9\) seconds is approximately \(2.27\) meters outward.

Example 10

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 10)

A mass of \(25\) kg is attached to a certain spring such that \(27\) Newtons of force is required to compress the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.1\) seconds.

Answer.

The IVP is given by

\[ 25x''+9x=0 \hspace{3em} x(0)=-3, x'(0)=0 \]

The position after \(9.1\) seconds is approximately \(2.04\) meters inward.

Example 11

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 11)

A mass of \(9\) kg is attached to a certain spring such that \(8\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.4\) seconds.

Answer.

The IVP is given by

\[ 9x''+4x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(2.4\) seconds is approximately \(0.0583\) meters outward.

Example 12

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 12)

A mass of \(9\) kg is attached to a certain spring such that \(48\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(4.5\) seconds.

Answer.

The IVP is given by

\[ 9x''+16x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(4.5\) seconds is approximately \(2.88\) meters outward.

Example 13

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 13)

A mass of \(4\) kg is attached to a certain spring such that \(50\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.2\) seconds.

Answer.

The IVP is given by

\[ 4x''+25x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(2.2\) seconds is approximately \(1.42\) meters outward.

Example 14

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 14)

A mass of \(4\) kg is attached to a certain spring such that \(50\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.2\) seconds.

Answer.

The IVP is given by

\[ 4x''+25x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(6.2\) seconds is approximately \(1.96\) meters outward.

Example 15

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 15)

A mass of \(16\) kg is attached to a certain spring such that \(16\) Newtons of force is required to stretch the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.6\) seconds.

Answer.

The IVP is given by

\[ 16x''+4x=0 \hspace{3em} x(0)=4, x'(0)=0 \]

The position after \(2.6\) seconds is approximately \(1.07\) meters outward.

Example 16

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 16)

A mass of \(9\) kg is attached to a certain spring such that \(32\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.0\) seconds.

Answer.

The IVP is given by

\[ 9x''+16x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(1.0\) seconds is approximately \(0.470\) meters outward.

Example 17

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 17)

A mass of \(9\) kg is attached to a certain spring such that \(75\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.1\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(3.1\) seconds is approximately \(1.32\) meters outward.

Example 18

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 18)

A mass of \(25\) kg is attached to a certain spring such that \(36\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.8\) seconds.

Answer.

The IVP is given by

\[ 25x''+9x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(9.8\) seconds is approximately \(3.68\) meters inward.

Example 19

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 19)

A mass of \(9\) kg is attached to a certain spring such that \(125\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.3\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(3.3\) seconds is approximately \(3.54\) meters outward.

Example 20

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 20)

A mass of \(9\) kg is attached to a certain spring such that \(64\) Newtons of force is required to stretch the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(8.2\) seconds.

Answer.

The IVP is given by

\[ 9x''+16x=0 \hspace{3em} x(0)=4, x'(0)=0 \]

The position after \(8.2\) seconds is approximately \(0.248\) meters inward.

Example 21

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 21)

A mass of \(4\) kg is attached to a certain spring such that \(80\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.8\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(6.8\) seconds is approximately \(2.56\) meters inward.

Example 22

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 22)

A mass of \(9\) kg is attached to a certain spring such that \(8\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.7\) seconds.

Answer.

The IVP is given by

\[ 9x''+4x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(9.7\) seconds is approximately \(1.97\) meters outward.

Example 23

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 23)

A mass of \(4\) kg is attached to a certain spring such that \(45\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.1\) seconds.

Answer.

The IVP is given by

\[ 4x''+9x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(6.1\) seconds is approximately \(4.81\) meters outward.

Example 24

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 24)

A mass of \(9\) kg is attached to a certain spring such that \(75\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(8.4\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(8.4\) seconds is approximately \(0.410\) meters outward.

Example 25

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 25)

A mass of \(9\) kg is attached to a certain spring such that \(125\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(4.3\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(4.3\) seconds is approximately \(3.17\) meters outward.

Example 26

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 26)

A mass of \(25\) kg is attached to a certain spring such that \(32\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(5.9\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(5.9\) seconds is approximately \(0.0157\) meters outward.

Example 27

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 27)

A mass of \(4\) kg is attached to a certain spring such that \(125\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.6\) seconds.

Answer.

The IVP is given by

\[ 4x''+25x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(1.6\) seconds is approximately \(3.27\) meters outward.

Example 28

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 28)

A mass of \(9\) kg is attached to a certain spring such that \(125\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.7\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(2.7\) seconds is approximately \(1.05\) meters outward.

Example 29

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 29)

A mass of \(25\) kg is attached to a certain spring such that \(48\) Newtons of force is required to compress the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.0\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=-3, x'(0)=0 \]

The position after \(6.0\) seconds is approximately \(0.262\) meters inward.

Example 30

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 30)

A mass of \(9\) kg is attached to a certain spring such that \(50\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.1\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(7.1\) seconds is approximately \(1.49\) meters outward.

Example 31

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 31)

A mass of \(4\) kg is attached to a certain spring such that \(64\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(8.8\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(8.8\) seconds is approximately \(1.26\) meters inward.

Example 32

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 32)

A mass of \(16\) kg is attached to a certain spring such that \(125\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.5\) seconds.

Answer.

The IVP is given by

\[ 16x''+25x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(9.5\) seconds is approximately \(3.85\) meters inward.

Example 33

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 33)

A mass of \(4\) kg is attached to a certain spring such that \(45\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.4\) seconds.

Answer.

The IVP is given by

\[ 4x''+9x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(1.4\) seconds is approximately \(2.52\) meters inward.

Example 34

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 34)

A mass of \(16\) kg is attached to a certain spring such that \(75\) Newtons of force is required to compress the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(8.7\) seconds.

Answer.

The IVP is given by

\[ 16x''+25x=0 \hspace{3em} x(0)=-3, x'(0)=0 \]

The position after \(8.7\) seconds is approximately \(0.361\) meters outward.

Example 35

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 35)

A mass of \(9\) kg is attached to a certain spring such that \(100\) Newtons of force is required to stretch the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.3\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=4, x'(0)=0 \]

The position after \(7.3\) seconds is approximately \(3.69\) meters outward.

Example 36

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 36)

A mass of \(25\) kg is attached to a certain spring such that \(36\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(4.0\) seconds.

Answer.

The IVP is given by

\[ 25x''+9x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(4.0\) seconds is approximately \(2.95\) meters outward.

Example 37

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 37)

A mass of \(25\) kg is attached to a certain spring such that \(80\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(4.3\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(4.3\) seconds is approximately \(4.78\) meters outward.

Example 38

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 38)

A mass of \(4\) kg is attached to a certain spring such that \(32\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.3\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(1.3\) seconds is approximately \(1.71\) meters inward.

Example 39

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 39)

A mass of \(25\) kg is attached to a certain spring such that \(16\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.2\) seconds.

Answer.

The IVP is given by

\[ 25x''+4x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(1.2\) seconds is approximately \(3.55\) meters inward.

Example 40

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 40)

A mass of \(9\) kg is attached to a certain spring such that \(8\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.4\) seconds.

Answer.

The IVP is given by

\[ 9x''+4x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(6.4\) seconds is approximately \(0.862\) meters inward.

Example 41

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 41)

A mass of \(9\) kg is attached to a certain spring such that \(125\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(5.3\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(5.3\) seconds is approximately \(4.15\) meters outward.

Example 42

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 42)

A mass of \(9\) kg is attached to a certain spring such that \(125\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(4.2\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(4.2\) seconds is approximately \(3.77\) meters outward.

Example 43

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 43)

A mass of \(4\) kg is attached to a certain spring such that \(48\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(5.8\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(5.8\) seconds is approximately \(1.70\) meters outward.

Example 44

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 44)

A mass of \(4\) kg is attached to a certain spring such that \(48\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.3\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(1.3\) seconds is approximately \(2.57\) meters inward.

Example 45

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 45)

A mass of \(25\) kg is attached to a certain spring such that \(32\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.2\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(2.2\) seconds is approximately \(0.376\) meters inward.

Example 46

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 46)

A mass of \(9\) kg is attached to a certain spring such that \(75\) Newtons of force is required to compress the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.2\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=-3, x'(0)=0 \]

The position after \(9.2\) seconds is approximately \(2.79\) meters outward.

Example 47

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 47)

A mass of \(25\) kg is attached to a certain spring such that \(18\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.6\) seconds.

Answer.

The IVP is given by

\[ 25x''+9x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(1.6\) seconds is approximately \(1.15\) meters inward.

Example 48

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 48)

A mass of \(4\) kg is attached to a certain spring such that \(64\) Newtons of force is required to stretch the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.6\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=4, x'(0)=0 \]

The position after \(7.6\) seconds is approximately \(3.50\) meters inward.

Example 49

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 49)

A mass of \(9\) kg is attached to a certain spring such that \(8\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.0\) seconds.

Answer.

The IVP is given by

\[ 9x''+4x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(2.0\) seconds is approximately \(0.470\) meters inward.

Example 50

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 50)

A mass of \(16\) kg is attached to a certain spring such that \(20\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(4.6\) seconds.

Answer.

The IVP is given by

\[ 16x''+4x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(4.6\) seconds is approximately \(3.33\) meters inward.

Example 51

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 51)

A mass of \(9\) kg is attached to a certain spring such that \(64\) Newtons of force is required to stretch the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.2\) seconds.

Answer.

The IVP is given by

\[ 9x''+16x=0 \hspace{3em} x(0)=4, x'(0)=0 \]

The position after \(9.2\) seconds is approximately \(3.82\) meters outward.

Example 52

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 52)

A mass of \(9\) kg is attached to a certain spring such that \(100\) Newtons of force is required to stretch the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.0\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=4, x'(0)=0 \]

The position after \(6.0\) seconds is approximately \(3.36\) meters inward.

Example 53

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 53)

A mass of \(16\) kg is attached to a certain spring such that \(50\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.6\) seconds.

Answer.

The IVP is given by

\[ 16x''+25x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(1.6\) seconds is approximately \(0.832\) meters inward.

Example 54

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 54)

A mass of \(16\) kg is attached to a certain spring such that \(50\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.1\) seconds.

Answer.

The IVP is given by

\[ 16x''+25x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(6.1\) seconds is approximately \(0.454\) meters outward.

Example 55

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 55)

A mass of \(25\) kg is attached to a certain spring such that \(12\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(5.4\) seconds.

Answer.

The IVP is given by

\[ 25x''+4x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(5.4\) seconds is approximately \(1.67\) meters inward.

Example 56

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 56)

A mass of \(16\) kg is attached to a certain spring such that \(16\) Newtons of force is required to stretch the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.8\) seconds.

Answer.

The IVP is given by

\[ 16x''+4x=0 \hspace{3em} x(0)=4, x'(0)=0 \]

The position after \(3.8\) seconds is approximately \(1.29\) meters inward.

Example 57

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 57)

A mass of \(16\) kg is attached to a certain spring such that \(20\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.4\) seconds.

Answer.

The IVP is given by

\[ 16x''+4x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(6.4\) seconds is approximately \(4.99\) meters inward.

Example 58

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 58)

A mass of \(16\) kg is attached to a certain spring such that \(27\) Newtons of force is required to compress the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.6\) seconds.

Answer.

The IVP is given by

\[ 16x''+9x=0 \hspace{3em} x(0)=-3, x'(0)=0 \]

The position after \(3.6\) seconds is approximately \(2.71\) meters outward.

Example 59

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 59)

A mass of \(4\) kg is attached to a certain spring such that \(75\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.8\) seconds.

Answer.

The IVP is given by

\[ 4x''+25x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(3.8\) seconds is approximately \(2.99\) meters inward.

Example 60

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 60)

A mass of \(9\) kg is attached to a certain spring such that \(12\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.8\) seconds.

Answer.

The IVP is given by

\[ 9x''+4x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(9.8\) seconds is approximately \(2.91\) meters outward.

Example 61

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 61)

A mass of \(4\) kg is attached to a certain spring such that \(75\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.6\) seconds.

Answer.

The IVP is given by

\[ 4x''+25x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(7.6\) seconds is approximately \(2.97\) meters outward.

Example 62

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 62)

A mass of \(4\) kg is attached to a certain spring such that \(27\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.6\) seconds.

Answer.

The IVP is given by

\[ 4x''+9x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(3.6\) seconds is approximately \(1.90\) meters outward.

Example 63

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 63)

A mass of \(25\) kg is attached to a certain spring such that \(12\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.8\) seconds.

Answer.

The IVP is given by

\[ 25x''+4x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(3.8\) seconds is approximately \(0.152\) meters outward.

Example 64

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 64)

A mass of \(25\) kg is attached to a certain spring such that \(32\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(4.6\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(4.6\) seconds is approximately \(1.72\) meters inward.

Example 65

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 65)

A mass of \(25\) kg is attached to a certain spring such that \(36\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.8\) seconds.

Answer.

The IVP is given by

\[ 25x''+9x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(9.8\) seconds is approximately \(3.68\) meters inward.

Example 66

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 66)

A mass of \(9\) kg is attached to a certain spring such that \(20\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.1\) seconds.

Answer.

The IVP is given by

\[ 9x''+4x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(2.1\) seconds is approximately \(0.850\) meters inward.

Example 67

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 67)

A mass of \(25\) kg is attached to a certain spring such that \(20\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.4\) seconds.

Answer.

The IVP is given by

\[ 25x''+4x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(7.4\) seconds is approximately \(4.92\) meters outward.

Example 68

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 68)

A mass of \(4\) kg is attached to a certain spring such that \(125\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.0\) seconds.

Answer.

The IVP is given by

\[ 4x''+25x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(3.0\) seconds is approximately \(1.73\) meters outward.

Example 69

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 69)

A mass of \(9\) kg is attached to a certain spring such that \(48\) Newtons of force is required to compress the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(5.5\) seconds.

Answer.

The IVP is given by

\[ 9x''+16x=0 \hspace{3em} x(0)=-3, x'(0)=0 \]

The position after \(5.5\) seconds is approximately \(1.49\) meters inward.

Example 70

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 70)

A mass of \(25\) kg is attached to a certain spring such that \(18\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.4\) seconds.

Answer.

The IVP is given by

\[ 25x''+9x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(1.4\) seconds is approximately \(1.33\) meters outward.

Example 71

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 71)

A mass of \(25\) kg is attached to a certain spring such that \(45\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.5\) seconds.

Answer.

The IVP is given by

\[ 25x''+9x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(3.5\) seconds is approximately \(2.52\) meters outward.

Example 72

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 72)

A mass of \(25\) kg is attached to a certain spring such that \(32\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(4.6\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(4.6\) seconds is approximately \(1.72\) meters outward.

Example 73

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 73)

A mass of \(16\) kg is attached to a certain spring such that \(36\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.9\) seconds.

Answer.

The IVP is given by

\[ 16x''+9x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(7.9\) seconds is approximately \(3.75\) meters inward.

Example 74

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 74)

A mass of \(4\) kg is attached to a certain spring such that \(64\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(8.5\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(8.5\) seconds is approximately \(1.10\) meters outward.

Example 75

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 75)

A mass of \(9\) kg is attached to a certain spring such that \(75\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.2\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(6.2\) seconds is approximately \(1.85\) meters inward.

Example 76

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 76)

A mass of \(25\) kg is attached to a certain spring such that \(80\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(5.1\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(5.1\) seconds is approximately \(2.96\) meters inward.

Example 77

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 77)

A mass of \(25\) kg is attached to a certain spring such that \(32\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.4\) seconds.

Answer.

The IVP is given by

\[ 25x''+16x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(2.4\) seconds is approximately \(0.684\) meters outward.

Example 78

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 78)

A mass of \(4\) kg is attached to a certain spring such that \(100\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.8\) seconds.

Answer.

The IVP is given by

\[ 4x''+25x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(2.8\) seconds is approximately \(3.02\) meters inward.

Example 79

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 79)

A mass of \(9\) kg is attached to a certain spring such that \(16\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.0\) seconds.

Answer.

The IVP is given by

\[ 9x''+4x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(3.0\) seconds is approximately \(1.66\) meters outward.

Example 80

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 80)

A mass of \(25\) kg is attached to a certain spring such that \(12\) Newtons of force is required to compress the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.1\) seconds.

Answer.

The IVP is given by

\[ 25x''+4x=0 \hspace{3em} x(0)=-3, x'(0)=0 \]

The position after \(9.1\) seconds is approximately \(2.64\) meters outward.

Example 81

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 81)

A mass of \(25\) kg is attached to a certain spring such that \(16\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.4\) seconds.

Answer.

The IVP is given by

\[ 25x''+4x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(1.4\) seconds is approximately \(3.39\) meters inward.

Example 82

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 82)

A mass of \(25\) kg is attached to a certain spring such that \(8\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.5\) seconds.

Answer.

The IVP is given by

\[ 25x''+4x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(6.5\) seconds is approximately \(1.71\) meters outward.

Example 83

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 83)

A mass of \(4\) kg is attached to a certain spring such that \(27\) Newtons of force is required to compress the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.2\) seconds.

Answer.

The IVP is given by

\[ 4x''+9x=0 \hspace{3em} x(0)=-3, x'(0)=0 \]

The position after \(2.2\) seconds is approximately \(2.96\) meters outward.

Example 84

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 84)

A mass of \(16\) kg is attached to a certain spring such that \(36\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.3\) seconds.

Answer.

The IVP is given by

\[ 16x''+9x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(2.3\) seconds is approximately \(0.614\) meters outward.

Example 85

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 85)

A mass of \(9\) kg is attached to a certain spring such that \(32\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(8.3\) seconds.

Answer.

The IVP is given by

\[ 9x''+16x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(8.3\) seconds is approximately \(0.142\) meters outward.

Example 86

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 86)

A mass of \(9\) kg is attached to a certain spring such that \(12\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.7\) seconds.

Answer.

The IVP is given by

\[ 9x''+4x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(7.7\) seconds is approximately \(1.23\) meters outward.

Example 87

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 87)

A mass of \(9\) kg is attached to a certain spring such that \(125\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.8\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(9.8\) seconds is approximately \(4.05\) meters inward.

Example 88

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 88)

A mass of \(25\) kg is attached to a certain spring such that \(12\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.6\) seconds.

Answer.

The IVP is given by

\[ 25x''+4x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(7.6\) seconds is approximately \(2.98\) meters inward.

Example 89

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 89)

A mass of \(25\) kg is attached to a certain spring such that \(16\) Newtons of force is required to stretch the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.2\) seconds.

Answer.

The IVP is given by

\[ 25x''+4x=0 \hspace{3em} x(0)=4, x'(0)=0 \]

The position after \(2.2\) seconds is approximately \(2.55\) meters outward.

Example 90

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 90)

A mass of \(9\) kg is attached to a certain spring such that \(50\) Newtons of force is required to stretch the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(3.5\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=2, x'(0)=0 \]

The position after \(3.5\) seconds is approximately \(1.80\) meters outward.

Example 91

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 91)

A mass of \(16\) kg is attached to a certain spring such that \(16\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(8.5\) seconds.

Answer.

The IVP is given by

\[ 16x''+4x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(8.5\) seconds is approximately \(1.78\) meters outward.

Example 92

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 92)

A mass of \(4\) kg is attached to a certain spring such that \(32\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(8.8\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(8.8\) seconds is approximately \(0.631\) meters inward.

Example 93

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 93)

A mass of \(4\) kg is attached to a certain spring such that \(64\) Newtons of force is required to compress the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.2\) seconds.

Answer.

The IVP is given by

\[ 4x''+16x=0 \hspace{3em} x(0)=-4, x'(0)=0 \]

The position after \(2.2\) seconds is approximately \(1.23\) meters outward.

Example 94

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 94)

A mass of \(16\) kg is attached to a certain spring such that \(16\) Newtons of force is required to stretch the mass \(4\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(2.9\) seconds.

Answer.

The IVP is given by

\[ 16x''+4x=0 \hspace{3em} x(0)=4, x'(0)=0 \]

The position after \(2.9\) seconds is approximately \(0.482\) meters outward.

Example 95

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 95)

A mass of \(9\) kg is attached to a certain spring such that \(125\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.1\) seconds.

Answer.

The IVP is given by

\[ 9x''+25x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(6.1\) seconds is approximately \(3.69\) meters outward.

Example 96

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 96)

A mass of \(16\) kg is attached to a certain spring such that \(45\) Newtons of force is required to compress the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(7.8\) seconds.

Answer.

The IVP is given by

\[ 16x''+9x=0 \hspace{3em} x(0)=-5, x'(0)=0 \]

The position after \(7.8\) seconds is approximately \(4.54\) meters inward.

Example 97

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 97)

A mass of \(4\) kg is attached to a certain spring such that \(45\) Newtons of force is required to stretch the mass \(5\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(6.2\) seconds.

Answer.

The IVP is given by

\[ 4x''+9x=0 \hspace{3em} x(0)=5, x'(0)=0 \]

The position after \(6.2\) seconds is approximately \(4.96\) meters inward.

Example 98

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 98)

A mass of \(4\) kg is attached to a certain spring such that \(27\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.5\) seconds.

Answer.

The IVP is given by

\[ 4x''+9x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(1.5\) seconds is approximately \(1.88\) meters inward.

Example 99

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 99)

A mass of \(4\) kg is attached to a certain spring such that \(27\) Newtons of force is required to stretch the mass \(3\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(9.1\) seconds.

Answer.

The IVP is given by

\[ 4x''+9x=0 \hspace{3em} x(0)=3, x'(0)=0 \]

The position after \(9.1\) seconds is approximately \(1.40\) meters outward.

Example 100

C7m - Mass-spring systems. Model and analyze mechanical oscillators with a second-order IVP. (ver. 100)

A mass of \(9\) kg is attached to a certain spring such that \(32\) Newtons of force is required to compress the mass \(2\) meters from its natural position.

Write an initial value problem (IVP) modeling the position of this mass when released from rest. Then solve this IVP to compute the mass's position after \(1.3\) seconds.

Answer.

The IVP is given by

\[ 9x''+16x=0 \hspace{3em} x(0)=-2, x'(0)=0 \]

The position after \(1.3\) seconds is approximately \(0.324\) meters outward.