Question 88: Gravitation

Question

A light planet is revolving around a massive star in a circular orbit of radius $\mathrm{R}$ with a period of revolution T. If the force of attraction between planet and star is proportional to $\mathrm{R}^{-3 / 2}$ then choose the correct option :

Accepted Answer

To find the correct option for the relationship between the period of revolution T and the radius of the orbit R, we will consider the force of attraction and its proportionality to $\mathrm{R}^{-3 / 2}$.
According to Newton's law of universal gravitation, the force of attraction $F$ between two masses $m_1$ and $m_2$ separated by a distance $r$ is given by
$$ F = \frac{G m_1 m_2}{r^2}, $$
where $G$ is the gravitational constant.
However, in this particular case, the force of attraction is given to be proportional to $\mathrm{R}^{-3 / 2}$, so we can write
$$ F \propto \frac{1}{R^{3/2}}. $$
Since the planet is in a circular orbit around the star, the centripetal force required to keep the planet in orbit must be provided by this gravitational force. Hence, we can write that
$$ \frac{m v^2}{R} \propto \frac{1}{R^{3/2}}, $$
where $m$ is the mass of the planet and $v$ is its orbital speed.
Simplifying this, we get
$$ v^2 \propto \frac{1}{R^{1/2}}. $$
Now, the speed $v$ can be related to the period T through the circumference of the orbit, which is given by $2\pi R$. The orbital speed is the circumference divided by the period:
$$ v = \frac{2\pi R}{T}. $$
Substituting this into our proportionality, we get
$$ \left( \frac{2\pi R}{T} \right)^2 \propto \frac{1}{R^{1/2}}, $$
which simplifies to
$$ \frac{4\pi^2 R^2}{T^2} \propto \frac{1}{R^{1/2}}. $$
Solving for $T^2$, we get
$$ T^2 \propto \frac{R^{2 + 1/2}}{4\pi^2}, $$
so
$$ T^2 \propto R^{5/2}. $$
Therefore, the correct option is 
Option C
$$ T^2 \propto R^{5/2}. $$

Answer

$\mathrm{T}^2 \propto \mathrm{R}^{7 / 2}$

Answer

$\mathrm{T}^2 \propto \mathrm{R}^3$

Answer

$\mathrm{T}^2 \propto \mathrm{R}^{5 / 2}$

Answer

$\mathrm{T}^2 \propto \mathrm{R}^{3 / 2}$

Gravitation

A light planet is revolving around a massive star in a circular orbit of radius \mathrm{R} with a period of revolution T. If the force of attraction between planet and star is proportional to \mathrm{R}^{-3 / 2} then choose the correct option :

Choose the correct answer: