## 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)

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.

#### 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)

#### 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.

#### 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.

#### 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)

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.