Chapter 4.4, Problem 101E

a.

To determine

Find the displacement when $t=0$

Answer to Problem 101E

  $\boxed{2}$

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 $y\left(t\right)2e^{-t}\cos 6t$ where $y$ is the displacement (in centimetres) and $t$ is the time (in seconds). Find the displacement when

  $t=0$

Calculation:

Consider that in Harmonic Motion the displacement from equilibrium of an oscillating weight suspended by a spring and subjrct to the damping effect of friction is given by

  $y\left(t\right)2e^{-t}\cos 6t$

Here $y$ is the displacement (in centimetres) and time (in seconds)

Consider the following figure

  

Substitute $t=0$ in the equation $y\left(t\right)2e^{-t}\cos 6t$

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

Because $a^0=1$ and trigonometric values $\cos 0=1$

Hence, displacement is $\boxed{2}$ cm

b.

To determine

Find the displacement when $t=\frac{1}{4}$.

Answer to Problem 101E

  $\boxed{0.11}cm$

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 $y\left(t\right)2e^{-t}\cos 6t$ where $y$ is the displacement (in centimetres) and $t$ is the time (in seconds). Find the displacement when

  $t=\frac{1}{4}$

Calculation:

Consider that in Harmonic Motion the displacement from equilibrium of an oscillating weight suspended by a spring and subjrct to the damping effect of friction is given by

  $y\left(t\right)2e^{-t}\cos 6t$

Here $y$ is the displacement (in centimetres) and time (in seconds)

Consider the following figure

  Substitute $t=\frac{1}{4}$ in the equation $y\left(t\right)2e^{-t}\cos 6t$

  $y\left(\frac{1}{4}\right)2e^{-\frac{1}{4}}\cos 6\frac{1}{4}$

  $\begin{array}{l}=2e^{\frac{1}{4}}\cos \left(\frac{3}{2}\right)\\ 2\times 0.7788\times 0.0707\end{array}$

Evaluate value of cosine function using TI-83 calculator as shown below,

First set the calculator at radian mode then enter the function as below,

  

And

  

Hence,

  $y\left(\frac{1}{4}\right)=2\times 0.7788\times 0.0707$

  $\begin{array}{l}=0.1101\\ \approx 0.11\end{array}$

Hence, the displacement is $\boxed{0.11}cm$

c.

To determine

Find the displacement when $t=\frac{1}{2}$.

Answer to Problem 101E

  $\boxed{-0.12cm}$

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 $y\left(t\right)2e^{-t}\cos 6t$ where $y$ is the displacement (in centimetres) and $t$ is the time (in seconds). Find the displacement when

  $t=\frac{1}{2}$

Calculation:

Consider that in Harmonic Motion the displacement from equilibrium of an oscillating weight suspended by a spring and subjrct to the damping effect of friction is given by

  $y\left(t\right)2e^{-t}\cos 6t$

Here $y$ is the displacement (in centimetres) and time (in seconds)

Consider the following figure

  

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

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

  $\begin{array}{l}=2e^{\frac{1}{2}}\cos \left(3\right)\\ 2\times 0.6065\times -0.9899\end{array}$

Evaluate value of cosine function using TI-83 calculator as shown below,

First set the calculator at radian mode then enter the function as below,

  

And

  

Hence,

  $\begin{array}{l}y\left(\frac{1}{2}\right)=2\times 0.6065\times -0.9899\\ =-1.200\end{array}$

Hence, the displacemnet is $\boxed{-0.12cm}$