Question

A laser beam is directed at the Moon, 380,000 km from Earth. The beam diverges at an angle \({\bf{\theta }}\) (Fig. 8–40) 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

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

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.

Given information

Given data:

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

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

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 \(5.3 \times {10^3}\;{\rm{m}}\).