Question

If Earth completes one revolution in $24$ hours, what is the angular displacement made by the Earth in $1$ hour? Express your answer in both radian and degree.The Moon is orbiting the Earth approximately once in $27$ days, what is the angle inverted by the Moon per day?

Answer 1

Hint:The orbit of the Earth around the Sun is considered to be circular. In terms of angle we have to calculate the amount of angle that it traversed in 24 hours and then in 1 hour. For the next question, we will use the same concept as used in the first question, to have a solution.

Complete step by step answer:
The path around which the Earth revolves around the Sun or the orbit of the Earth is considered to be circular. The total angle in a circle measured from its centre is ${360^ \circ }$. It means that in a revolution the Earth completes a total angular displacement of ${360^ \circ }$.According to the given question, we get to know that in $24$ hours, the Earth completes ${360^ \circ }$.

Thus, from quadratic formula in $1$ hour, the angular displacement traversed by it is,
$\dfrac{{{{360}^ \circ }}}{{24}} = {15^ \circ }$.
In terms of radian we get,
${180^ \circ } = {\pi ^c}$
Again by quadratic relation we get,
${1^ \circ } = \dfrac{\pi }{{180}}$
$ \Rightarrow {15^ \circ } = {\left( {\dfrac{\pi }{{12}}} \right)^c}$
The angular displacement made by the Earth in $1$ hour is ${15^ \circ }$ in the degree system and $\dfrac{\pi }{{12}}$ in the radian system.Similarly, the Moon traverses the Earth in $27$ days.Therefore, in $27$ days the moon covers ${360^ \circ }$.

In $1$ day the moon covers,
$\dfrac{{{{360}^ \circ }}}{{27}} = {\left( {\dfrac{{40}}{3}} \right)^ \circ }$
In radian system we get,
${\left( {\dfrac{{40}}{3}} \right)^ \circ } = \left( {\dfrac{{40}}{3}} \right) \times \dfrac{\pi }{{180}} \\
\therefore {\left( {\dfrac{{40}}{3}} \right)^ \circ } = \dfrac{{2\pi }}{{27}}$

Thus, the moon per day travels, ${\left( {\dfrac{{40}}{3}} \right)^ \circ }$ in degree system and $\dfrac{{2\pi }}{{27}}$ in radian system.

Note:It must be noted that the orbit of the Earth around the Sun is not circular. It is elliptical in shape, but due to the efficiency in mathematical approach we consider it to be circular. Multiplying any degree with $\dfrac{\pi }{{180}}$ we get the value in the radian system.