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Chapter 8: 8-3P (page 198)

A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges at an angle \({\bf{\theta }}\) (Fig. 8鈥�40) of \({\bf{1}}{\bf{.4 \times 1}}{{\bf{0}}^{{\bf{ - 5}}}}\;{\bf{rad}}\). What diameter spot will it make on the Moon?

FIGURE 8-40 Problem 3.

Short Answer

Expert verified

The diameter of the circular spot that formed on the surface of the Moon is \(5.3 \times {10^3}\;{\rm{m}}\).

Step by step solution

01

Meaning of angular motion

The angular motion includes the circular or rotational motion of an object. The circular motion can be estimated using linear units or angular units.

The angular units refer to revolutions, radians, and degrees.

02

Given information

Given data:

The distance between the Earth and the Moon is \(r = 380000\;{\rm{km}}\).

The diverging angle is \(\theta = 1.4 \times {10^{ - 5}}\;{\rm{rad}}\).

03

Calculate the diameter of the spot

The diameter of the spot can be calculated as:

\(\begin{aligned}{l}d &= r\theta \\d &= \left( {\left( {380000\;{\rm{km}}} \right)\left( {\frac{{{\rm{1}}{{\rm{0}}^{\rm{3}}}\;{\rm{m}}}}{{{\rm{1}}\;{\rm{km}}}}} \right)} \right)\left( {1.4 \times {{10}^{ - 5}}\;{\rm{rad}}} \right)\\d &= 5.3 \times {10^3}\;{\rm{m}}\end{aligned}\)

Thus, the diameter of the circular spot that formed on the surface of the Moon is \(5.3 \times {10^3}\;{\rm{m}}\).

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Most popular questions from this chapter

(a) A yo-yo is made of two solid cylindrical disks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concentric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.013 m. Use conservation of energy to calculate the linear speed of the yo-yo just before it reaches the end of its 1.0-m-long string, if it is released from rest. (b) What fraction of its kinetic energy is rotational?

I) Water drives a waterwheel (or turbine) of radius R = 3.0 m as shown in Fig. 8鈥�66. The water enters at a speed\({v_1} = 7.0\;{\rm{m/s}}\)and exits from the waterwheel at a speed\({v_2} = 3.8\;{\rm{m/s}}\). (a) If 85 kg of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? (b) What is the torque the water applies to the waterwheel? (c) If the water causes the waterwheel to make one revolution every 5.5 s, how much power is delivered to the wheel?

An oxygen molecule consists of two oxygen atoms whose total mass is \({\bf{5}}{\bf{.3 \times 1}}{{\bf{0}}^{{\bf{ - 26}}}}\;{\bf{kg}}\) and the moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is \({\bf{1}}{\bf{.9 \times 1}}{{\bf{0}}^{{\bf{ - 46}}}}\;{\bf{kg}} \cdot {{\bf{m}}^{\bf{2}}}\). From these data, estimate the effective distance between the atoms.

Question:(II) A ball of radius rrolls on the inside of a track of radius R(see Fig. 8鈥�53). If the ball starts from rest at the vertical edge of the track, what will be its speed when it reaches the lowest point of the track, rolling without slipping?

Question: (III) An asteroid of mass \({\bf{1}}{\bf{.0 \times 1}}{{\bf{0}}^{\bf{5}}}\;{\bf{kg}}\) traveling at a speed of \({\bf{35}}\;{{{\bf{km}}} \mathord{\left/{\vphantom {{{\bf{km}}} {\bf{s}}}} \right.} {\bf{s}}}\) relative to the Earth, hits the Earth at the equator tangentially, in the direction of Earth鈥檚 rotation, and is embedded there. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.
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