Chapter 10, Problem 55A

A turntable that turns 10 revolutions each second is located on top of a mountain. Mounted on the turntable is a laser that emits a bright beam of light. As the turn-table and laser rotate, the beam also rotates and sweeps across the sky. On a dark night the beam reaches some clouds 10 km away.

a. How fast does the spot of laser light sweep across the clouds?

b. How fast does the spot of laser light sweep across clouds that are 20 km away?

c. At what distance will the laser beam sweep across the sky at the speed of light (300,000 km/s)?

To determine

The speed of the spot of laser light sweep across the clouds.

Answer to Problem 55A

  $628\text{km}/\text{s}$

Explanation of Solution

Given info:

The angular velocity of the turntable is $\omega =10\text{revolutions}/\text{s}$ .

The distance is $r=10\text{ km}$ .

Formula used:

The expression for the speed of the laser light as follows:

  $v=\frac{2\pi r}{t}$

Here, $r$ is the distance and $t$ is the time.

Calculation:

The time taken for one complete rotation as follows:

  $\begin{array}{l}t=\frac{2\pi }{\omega }\\ t=\frac{2\pi }{10\text{revolutions}/\text{s}\times \frac{2\pi \text{ rad}}{1\text{ revolutions}}}\\ t=0.1\text{ s}\end{array}$

The speed of the laser light as follows:

  $\begin{array}{l}v=\frac{2\pi r}{t}\\ v=\frac{2\pi \times \left(10\text{ km}\right)}{\left(0.1\text{ s}\right)}\\ v=628\text{km}/\text{s}\end{array}$

Conclusion:

Thus, the speed of the spot of laser light sweep across the clouds is $628\text{km}/\text{s}$ .

To determine

The speed of the spot of laser light sweep across the clouds.

Answer to Problem 55A

  $1,257\text{km}/\text{s}$

Explanation of Solution

Given info:

The angular velocity of the turntable is $\omega =10\text{revolutions}/\text{s}$ .

The distance is $r=20\text{ km}$ .

Formula used:

The expression for the speed of the laser light as follows:

  $v=\frac{2\pi r}{t}$

Here, $r$ is the distance and $t$ is the time.

Calculation:

Refer part (a).

The time taken for one complete rotation is

  $t=0.1\text{ s}$ .

The speed of the laser light as follows:

  $\begin{array}{l}v=\frac{2\pi r}{t}\\ v=\frac{2\pi \times \left(20\text{ km}\right)}{\left(0.1\text{ s}\right)}\\ v=1,257\text{km}/\text{s}\end{array}$

Conclusion:

Thus, the speed of the spot of laser light sweep across the clouds is $1,257\text{km}/\text{s}$ .

To determine

The distance of the laser beam sweep across the sky.

Answer to Problem 55A

  $4,775\text{ km}$

Explanation of Solution

Given info:

The angular velocity of the turntable is $\omega =10\text{revolutions}/\text{s}$ .

The speed of the spot of laser light sweep across the clouds is $v=300,000\text{km}/\text{s}$ .

Formula used:

The expression for the speed of the laser light as follows:

  $v=\frac{2\pi r}{t}$

Here, $r$ is the distance and $t$ is the time.

Calculation:

Refer part (a).

The time taken for one complete rotation is

  $t=0.1\text{ s}$ .

The distance of the laser beam sweep across the sky as follows:

  $\begin{array}{l}v=\frac{2\pi r}{t}\\ r=\frac{\left(0.1\text{ s}\right)\left(300,000\text{km}/\text{s}\right)}{2\pi }\\ r=4,775\text{ km}\end{array}$

Conclusion:

Thus, the distance of the laser beam sweep across the sky is $4,775\text{ km}$ .