Chapter 2, Problem 75RE

(a).

To determine

To find: The velocity of the light moving along the shore when it reaches point $A$

Answer to Problem 75RE

The velocity $-0.6km/\sec$

Explanation of Solution

Given:

  $\frac{d\theta }{dt}=-0.6\text{rad/sec},x=\tan \theta$ .

Concept Used:

Velocity is the derivative of the positive function with respect to time. At time $t$ the velocity is:

  $v\left(t\right)=\frac{ds}{dt}=\underset{\Delta t\to 0}{\lim }\frac{f\left(t+\Delta t\right)-f\left(t\right)}{\Delta t}$

Calculation:

  $\begin{array}{c}v=\frac{d}{dt}\left[x\right]\\ =\frac{d}{dt}(\tan \left(\theta \right))\\ ={\sec }^2\theta \frac{d\theta }{dt}\end{array}$

At point A , $x=0,\theta =0$

  $\begin{array}{c}{\left[\frac{dx}{dt}\right]}_{x=0,\theta =0}=1\times \left(-0.6\right)\\ =-0.6\text{km/sec}\end{array}$

Conclusion: The velocity $-0.6\text{km/sec}$ .

b.

To determine

To calculate: The revolution per minute.

Answer to Problem 75RE

The revolution per min is $=5.73\text{rpm}$

Explanation of Solution

Given:

  $\frac{d\theta }{dt}=-0.6\text{rad/sec},x=\tan \theta$

Calculation:

  $\begin{array}{c}\frac{0.6\text{rad}}{1\sec }\times \frac{1\text{rev}}{2\pi \text{rad}}\times \frac{60\sec }{1\min }=\frac{18}{\pi }\text{rev}/\min \\ =5.73\text{rpm}\end{array}$

Conclusion:

The revolution per min is $=5.73\text{rpm}$