Question

If the angular diameter of the Moon is $${30}^{{}^{\prime }}$$, how far from the eye must a coin of diameter $$2.2cm$$ be kept to hide the Moon?

Answer 1

Hint: Convert the given angle in degree to radians by multiplying it with $${\displaystyle \frac{\pi }{180}}$$. Now draw the figure representing the same. Take the distance at which the coin is placed as $$r$$. Substitute the values of angular diameter and arc length in the formula for angular displacement and get the value of $$r$$.
Complete step-by-step answer:
We have been given the angular diameter of the moon as $${30}^{{}^{\prime }}$$. Now we need to place a coin of $$2.2cm$$, at a particular distance from the eye such that we can hide the moon. Let us assume that the coin is kept at a distance of $$r$$ from the eye to hide the moon completely.
Let us consider the eye of the observer to be $$E$$. Let AB be the diameter of the coin, which we have been given as $$2.2cm$$. Thus,
$$AB=2.2cm$$
Let us draw a figure representing the same as we have told above.

Now from the figure let us consider arc APB as equal to the diameter AB of the coin. Hence we can say that,
$$arcAPB=AB=2.2cm$$
Now let us consider the angular diameter of the moon as, $$\theta$$.
Thus, $$\theta ={30}^{{}^{\prime }}$$.
Now let us convert the given angle of degrees to radians. To convert degrees to radians multiply the angle with $${\displaystyle \frac{\pi }{180}}$$. Let us consider the angle as equal to $$\theta$$. Hence, $$\theta =30{}^{\circ }$$. Now let us convert it to radians.
$$\begin{array}{rl}& \theta ={30}^{{}^{\prime }}=({\displaystyle \frac{30}{60}}\times {\displaystyle \frac{\pi }{180}})\\ & \theta ={\displaystyle \frac{1}{2}}\times {\displaystyle \frac{\pi }{180}}={\displaystyle \frac{\pi }{360}}\end{array}$$
Thus we got the angular displacement, $$\theta ={\displaystyle \frac{\pi }{360}}$$
We know the basic formula that,
Angular displacement, $$\theta ={\displaystyle \frac{arc}{radius}}$$
We got $$\theta ={\displaystyle \frac{\pi }{360}}$$ and we know arc length $$2.2cm$$, which is the diameter of the coin. Let us take the radius as $$r$$. Now let us substitute all these values in the formula for angular displacement.
$$\begin{array}{rl}& \theta ={\displaystyle \frac{arc}{radius}}\\ & {\displaystyle \frac{\pi }{360}}={\displaystyle \frac{2.2}{r}}\end{array}$$
Now apply cross multiplication property in the above expression and simplify it.
$$\begin{array}{rl}& r={\displaystyle \frac{2.2\times 360}{\pi }}\\ & \pi ={\displaystyle \frac{22}{7}}\\ & r={\displaystyle \frac{2.2\times 360\times 7}{22}}=0.1\times 360\times 7\\ & r=0.1\times 360\times 7=252\\ & r=252\end{array}$$
Thus we got the distance as $$252cm$$.
Hence, the coin should be kept at a distance of $$252cm$$ from the eye of the observer.

Note: Don’t forget to convert the given angle into radians, as the angular displacement of an object is always given in radians. Angular displacement is the shortest angle between the initial and final position for a given object, which is having a circular motion.
Students often make mistakes by directly multiplying $${30}^{\prime }$$ by $$\left({\displaystyle \frac{\pi }{180}}\right)$$, and forget to divide it by 60.