Question

If the radius of earth’s orbit is made ¼ the duration of the year will become
A. 8 times
B. 4 times
C. 1/8 times
D. 1/4 times

Answer 1

The duration of a year is basically the time required for planet earth to complete one complete revolution around the Sun. Here, we are dealing with the motion of planets. So, we shall be using Kepler’s laws of planetary motion. Johannes Kepler, a German astrologer gave three laws defining the motion of planets. We shall be using the third law of Kepler which states that the square of time required for one complete revolution of a planet is directly proportional to the cube of the radius of its orbit.

Formula Used:
${{T}^{2}}\propto {{r}^{3}}$

Complete step by step answer:
Since we are dealing with planetary motion, we will be using Kepler’s laws. We are asked to find the relation between the radius of orbit and time period of one revolution. Therefore, we will be using third law of Kepler which provides us this relation
Therefore, let us say that the time period of one revolution of earth around the sun is T and the radius of earth’s orbit is R. We have been asked to calculate the time period say T’ if the radius is made 1/4 th of the original radius i.e. ${{r}_{2}}=\dfrac{R}{4}$
${{T}^{2}}\propto {{r}^{3}}$
After substituting the values in above equation
We get,
${{T}^{2}}=k{{R}^{3}}$
Where, k is any assumed constant for proportionality.
Therefore,
$k=\dfrac{{{T}^{2}}}{{{R}^{3}}}$ …………….. (1)
Similarly, for the second instance when radius ${{r}_{2}}=\dfrac{R}{4}$ and T = T’
We get,
$T{{'}^{2}}=k{{\left( \dfrac{R}{4} \right)}^{3}}$
Therefore,
$k=\dfrac{64T{{'}^{2}}}{{{R}^{3}}}$ ……………….. (2)
Therefore, from (1) and (2)
We get,
$\dfrac{{{T}^{2}}}{{{R}^{3}}}$ = $\dfrac{64T{{'}^{2}}}{{{R}^{3}}}$
On solving,
We get,
$\dfrac{T{{'}^{2}}}{{{T}^{2}}}=\dfrac{1}{64}$
Therefore,
$\dfrac{T'}{T}=\dfrac{1}{8}$
Therefore, if the radius is made 1/4 th of the original radius the period would be $\dfrac{1}{8}$ of the original time period.
Therefore, the correct answer is option C.

Note:
Kepler proposed three laws for motion of planets. The first law states that the orbit of every planet is ellipse with the Sun at the focus. The second law states that a line joining the sun and the planet sweeps out equal area in equal time. The third law states that the square of time period is directly proportional to the cube of the radius of orbit of the planet.