Chapter 16, Problem 71P

(a)

To determine

The maximum displacement and maximum speed of a point on the string a $x=0.10\text{m}$ .

Explanation of Solution

Given:

The wave function of standing wave is given as $y\left(x,t\right)=\left(0.020\right)\sin \left(4\pi x\right)\cos \left(60\pi t\right)$ .

Formula used:

Write expression for wave function as maximum displacement.

  $y_{\max }\left(x\right)=\left(0.020\right)\sin \left(4\pi x\right)$........ (1)

Differentiate above expression for $t$ .

  $v_{y\max }\left(x\right)=\left(1.2\pi \text{m}/\text{s}\right)\sin \left[\left(4\pi {\text{m}}^{-1}\right)x\right]$....... (2)

Calculation:

Substitute $0.10\text{m}$ for $x$ in equation (1).

  $\begin{array}{l}{y}_{\max }\left(0.10\text{m}\right)=\left(0.020\right)\sin \left(4\pi \left(0.10\text{m}\right)\right)\\ y_{\max }\left(0.10\text{m}\right)=0.019\text{m}\end{array}$

Substitute $0.10\text{m}$ for $x$ in equation (2).

  $\begin{array}{l}{v}_{y\max }\left(0.10\text{m}\right)=\left(1.2\pi \text{m}/\text{s}\right)\sin \left[\left(4\pi {\text{m}}^{-1}\right)\left(0.10\text{m}\right)\right]\\ v_{y\max }\left(0.10\text{m}\right)=3.6\text{m}/\text{s}\end{array}$

Conclusion:

Thus, the maximum displacement is $0.019\text{m}$ and maximum speed is $3.6\text{m}/\text{s}$ .

(b)

To determine

The maximum displacement and maximum speed of a point on the string a $x=0.25\text{m}$ .

Explanation of Solution

Given:

The wave function of standing wave is given as $y\left(x,t\right)=\left(0.020\right)\sin \left(4\pi x\right)\cos \left(60\pi t\right)$ .

Formula used:

Write expression for wave function as maximum displacement.

  $y_{\max }\left(x\right)=\left(0.020\right)\sin \left(4\pi x\right)$........ (1)

Differentiate above expression for $t$ .

  $v_{y\max }\left(x\right)=\left(1.2\pi \text{m}/\text{s}\right)\sin \left[\left(4\pi {\text{m}}^{-1}\right)x\right]$....... (2)

Calculation:

Substitute $0.25\text{m}$ for $x$ in equation (1).

  $\begin{array}{l}{y}_{\max }\left(0.25\text{m}\right)=\left(0.020\right)\sin \left(4\pi \left(0.25\text{m}\right)\right)\\ y_{\max }\left(0.25\text{m}\right)=0\text{m}\end{array}$

Substitute $0.25\text{m}$ for $x$ in equation (2).

  $\begin{array}{l}{v}_{y\max }\left(0.25\text{m}\right)=\left(1.2\pi \text{m}/\text{s}\right)\sin \left[\left(4\pi {\text{m}}^{-1}\right)\left(0.25\text{m}\right)\right]\\ v_{y\max }\left(0.25\text{m}\right)=0\text{m}/\text{s}\end{array}$

Conclusion:

Thus, the maximum displacement is $0\text{m}$ and maximum speed is $0\text{m}/\text{s}$ .

(c)

To determine

The maximum displacement and maximum speed of a point on the string a $x=0.30\text{m}$ .

Explanation of Solution

Given:

The wave function of standing wave is given as $y\left(x,t\right)=\left(0.020\right)\sin \left(4\pi x\right)\cos \left(60\pi t\right)$ .

Formula used:

Write expression for wave function as maximum displacement.

  $y_{\max }\left(x\right)=\left(0.020\right)\sin \left(4\pi x\right)$........ (1)

Differentiate above expression for $t$ .

  $v_{y\max }\left(x\right)=\left(1.2\pi \text{m}/\text{s}\right)\sin \left[\left(4\pi {\text{m}}^{-1}\right)x\right]$....... (2)

Calculation:

Substitute $0.30\text{m}$ for $x$ in equation (1).

  $\begin{array}{l}{y}_{\max }\left(0.30\text{m}\right)=\left(0.020\right)\sin \left(4\pi \left(0.30\text{m}\right)\right)\\ y_{\max }\left(0.30\text{m}\right)=-0.011\text{m}\end{array}$

Substitute $0.30\text{m}$ for $x$ in equation (2).

  $\begin{array}{l}{v}_{y\max }\left(0.30\text{m}\right)=\left(1.2\pi \text{m}/\text{s}\right)\sin \left[\left(4\pi {\text{m}}^{-1}\right)\left(0.30\text{m}\right)\right]\\ v_{y\max }\left(0.30\text{m}\right)=2.2\text{m}/\text{s}\end{array}$

Conclusion:

Thus, the maximum displacement is $-0.011\text{m}$ and maximum speed is $2.2\text{m}/\text{s}$ .

(d)

To determine

The maximum displacement and maximum speed of a point on the string a $x=0.50\text{m}$ .

Explanation of Solution

Given:

The wave function of standing wave is given as $y\left(x,t\right)=\left(0.020\right)\sin \left(4\pi x\right)\cos \left(60\pi t\right)$ .

Formula used:

Write expression for wave function as maximum displacement.

  $y_{\max }\left(x\right)=\left(0.020\right)\sin \left(4\pi x\right)$........ (1)

Differentiate above expression for $t$ .

  $v_{y\max }\left(x\right)=\left(1.2\pi \text{m}/\text{s}\right)\sin \left[\left(4\pi {\text{m}}^{-1}\right)x\right]$....... (2)

Calculation:

Substitute $0.50\text{m}$ for $x$ in equation (1).

  $\begin{array}{l}{y}_{\max }\left(0.50\text{m}\right)=\left(0.020\right)\sin \left(4\pi \left(0.50\text{m}\right)\right)\\ y_{\max }\left(0.50\text{m}\right)=0\text{m}\end{array}$

Substitute $0.50\text{m}$ for $x$ in equation (2).

  $\begin{array}{l}{v}_{y\max }\left(0.50\text{m}\right)=\left(1.2\pi \text{m}/\text{s}\right)\sin \left[\left(4\pi {\text{m}}^{-1}\right)\left(0.50\text{m}\right)\right]\\ v_{y\max }\left(0.50\text{m}\right)=0\text{m}/\text{s}\end{array}$

Conclusion:

Thus, the maximum displacement is $0\text{m}$ and maximum speed is $0\text{m}/\text{s}$ .