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.