Question

The periods of two planets round the sun are in the ratio $1:8$ then their radii will be in the ratio:
A) $1:2$
B) $1:4$
C) $1:64$
D) None of the above

Answer 1

Hint Equate the gravitational force with the centripetal force. On simplifying we arrive at the equation ${r^3} = k{T^2}$ . Here, the cube of the radius is directly proportional to the square of the time period. Write the equation as ratio of two planets $\dfrac{{{r_1}}}{{{r_2}}} = {(\dfrac{{{T_1}}}{{{T_2}}})^{\dfrac{2}{3}}}$ . Substitute the ratio of time period and evaluate to find the ratio of the radii.

Complete step-by-step solution As we know that when 2 planets revolve around each other, they exert an equal and opposite gravitational force on each other. This gravitational force is given as:
$F = \dfrac{{GMm}}{{{r^2}}}$
This gravitational force is equal and opposite in direction to the centripetal force that exists between the 2 planets. This centripetal force is:
$F = m{\omega ^2}r$
Equating the 2 forces,
$m{\omega ^2}r = \dfrac{{GMm}}{{{r^2}}}$
Where $\omega = \dfrac{{2\pi }}{T}$
Solving the above equation, we get
${r^3} = k{T^2}$
Where k is a constant,
Therefore, the ratio of radii of the 2 planets is

$$\dfrac{{{r_1}}}{{{r_2}}} = {(\dfrac{{{T_1}}}{{{T_2}}})^{\dfrac{2}{3}}} \\\ \dfrac{{{r_1}}}{{{r_2}}} = {(\dfrac{1}{8})^{\dfrac{2}{3}}} \\\ \dfrac{{{r_1}}}{{{r_2}}} = \dfrac{1}{4} \\\$$

So, the correct answer is option B.

Note Here we need not consider the constant k as it is cancelled in the equation. The constant is the same for only a single pair of planets. Different combinations of planets will have different values of K.
According to Kepler's law of periods, the square of period of revolution of any planet around the sun is directly proportional to the cube of the semi-major axis of the orbit.
${T^2} \propto {a^3}$