Question

The earth takes 1 year to make a complete revolution around the sun whereas Pluto takes nearly 248 of our earth years to orbit the sun; what is the main reason for this?
(A) Pluto being small in size is slower than earth.
(B) Pluto being more massive moves more slowly.
(C )Strong winds on the Earth makes it move faster.
(D) Pluto being far from the sun has to travel more distance than the earth.

Answer 1

Hint: For a planet moving in a circular orbit the gravitational force and centripetal forces are equal. On equating gravitational force and centripetal force we get a new equation of velocity of the planet. We know that the time period is the ratio of circumference of circular orbit to the velocity of the planet. Thus substitute the value of velocity in the second equation, we will get the relation connecting time period and its radius. Then compare both the cases of earth and pluto.

Complete answer:
Gravitational force, $F=\dfrac{GMm}{{{R}^{2}}}$
Where, G is the gravitational constant.
M is the mass of earth.
m is the mass of the planet revolving around earth.
R is the radius of earth.
Centripetal force, $F=\dfrac{m{{v}^{2}}}{R}$
m is the mass of the planet revolving around earth.
R is the radius of earth.
v is the velocity of a planet revolving around the earth.
For a planet moving in a circular orbit the gravitational force and centripetal forces are equal.
Thus we get,
$\dfrac{GMm}{{{R}^{2}}}=\dfrac{m{{v}^{2}}}{R}$
$\Rightarrow $ $\dfrac{GM}{R}={{v}^{2}}$
$\Rightarrow v=\sqrt{\dfrac{GM}{R}}$ ……………..(1)
The time period, T is given by,
$T=\dfrac{2\pi R}{v}$ ……………….(2)
where, $2\pi R$ is the circumference of the circular orbit and v is the velocity.
Substituting equation (1) in equation (2),
$T=\dfrac{2\pi R}{\sqrt{\dfrac{GM}{R}}}$
$\Rightarrow T=2\pi R\sqrt{\dfrac{R}{GM}}$
$\Rightarrow T=2\pi \dfrac{{{R}^{\dfrac{3}{2}}}}{\sqrt{GM}}$
$\Rightarrow T\propto {{R}^{\dfrac{3}{2}}}$
This shows that the time period is proportional to ${{R}^{\dfrac{3}{2}}}$ .
So when earth takes 1 to complete revolution around the sun whereas Pluto takes nearly 248 of our earth years to orbit the sun to complete the revolution, because Pluto is small in size and slower than earth.

So, the correct answer is “Option A”.

Note:
For a planet moving in a circular orbit the gravitational force and centripetal forces are equal. The time period is the ratio of circumference of circular orbit to the velocity of the planet. The time period is proportional to ${{R}^{\dfrac{3}{2}}}$. So when earth takes 1 to complete revolution around the sun whereas Pluto takes nearly 248 of our earth years to orbit the sun to complete the revolution, because Pluto is small in size and slower than earth.