Chapter 4.2, Problem 50E

a.

To determine

Complete the table.

Answer to Problem 50E

  $\begin{array}{cccccc}t& 0& 1/4& 1/2& 3/4& 1\\ y& 0.5& 0.028& -0.3& -0.05& 0.177\end{array}$

Explanation of Solution

Given information:

The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by $y\left(t\right)=\frac{1}{2}{e}^{-t}\cos 6t$ where $y$ is the displacement in feet and $t$ is the time in seconds.

  $\begin{array}{cccccc}t& 0& 1/4& 1/2& 3/4& 1\\ y& & & & & \end{array}$

Complete the table.

Calculation:

Consider the given function,

  $y\left(t\right)=\frac{1}{2}{e}^{-t}\cos 6t$

To calculate displacement substitute $t=o$

  $\begin{array}{l}y\left(t\right)=\frac{1}{2}{e}^{-t}\cos 6t\\ y\left(0\right)=\frac{1}{2}{e}^{-0}\cos 6\times 0\\ y\left(0\right)=0.5\end{array}$

Hence displacement at $t=0$ is $0.5meters$ .

Now substitute $t=1/4$ .

  $\begin{array}{l}t=1/4\\ y\left(1/4\right)=\frac{1}{2}{e}^{-1/4}\cos 6\times 1/4\\ =0.028\end{array}$

Hence displacement at $t=1/4$ is $0.028meters$

Now substitute $t=1/2$ .

  $\begin{array}{l}y\left(1/2\right)=\frac{1}{2}{e}^{-1/2}\cos 6\times 1/2\\ =-0.3meters\end{array}$

Hence displacement at $t=1/2$ is $-0.3meters$

Now substitute $t=3/4$ .

  $\begin{array}{l}y\left(3/4\right)=\frac{1}{2}{e}^{-3/4}\cos 6\times 3/4\\ =-0.05meters\end{array}$ .

Hence displacement at $t=3/4$ is $-0.05meters$ .

Now substitute $t=1$ .

  $\begin{array}{l}y\left(1\right)=\frac{1}{2}{e}^{-1}\cos 6\times 1\\ =0.177meters\end{array}$

Hence displacement at $t=1$ is $0.177meters$ .

Hence the table is, $\begin{array}{cccccc}t& 0& 1/4& 1/2& 3/4& 1\\ y& 0.5& 0.028& -0.3& -0.05& 0.177\end{array}$ .

b.

To determine

To approximate the time when the weight reaches equilibrium.

Answer to Problem 50E

  $\boxed{5\sec }$

Explanation of Solution

Given information:

The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by $y\left(t\right)=\frac{1}{2}{e}^{-t}\cos 6t$ where $y$ is the displacement in feet and $t$ is the time in seconds.

  $\begin{array}{cccccc}t& 0& 1/4& 1/2& 3/4& 1\\ y& & & & & \end{array}$

Use the table feature of a graphing utility to approximate the time when the weight reaches equilibrium.

Calculation:

The system reaches equilibrium when the displacement is zero. so the approximate time, when the weight reaches equilibrium is $5\sec$

Hence, the time when weight reaches equilibrium is $\boxed{5\sec }$ .

c.

To determine

What appears to happen to the displacement as $t$ increases?

Answer to Problem 50E

  $\boxed{zero}$

Explanation of Solution

Given information:

The displacement from equilibrium of an oscillating weight suspended by a spring and subject to the damping effect of friction is given by $y\left(t\right)=\frac{1}{2}{e}^{-t}\cos 6t$ where $y$ is the displacement in feet and $t$ is the time in seconds.

  $\begin{array}{cccccc}t& 0& 1/4& 1/2& 3/4& 1\\ y& & & & & \end{array}$

What appears to happen to the displacement as $t$ increases?

Calculation:

As the time increase, the weight continues to oscillate but its displacement gets smaller, and finally approaches zero.

Hence, the displacement gets smaller, and finally approaches $\boxed{zero}$ .