Question

the average distance of uranus from sun is about 20 AU and that of pluto is about 40 AU.find the ratio of the time periods of revolution of these planets around the sun.

Answer 1

Law of Time Period:

=>Planets Orbiting around the sun have Time Periods such that :

$T^{2} \propto a^{3} : ---(1)$

where a is approximate distance  of planet from Sun. (in case of Circular Orbits) 

=>Proof:

We are considering Circular Orbit.

Let us consider a planet of mass 'm' orbiting(uniform acceleration) around a Sun of Mass M .

(M>>>m) .

Then we can say,

=>Gravitational Force should Balance Radial force

$\frac{GMm}{r^{2}} =m{\omega}^{2}r \\ \\ => {\omega}^{2}= \frac{GM}{r^{3}} --(1)$

Also ,we know that,

$T=\frac{2\pi}{\omega} --(2)$

Using eq. (1) and (2), we get

$T^{2} =\frac{4\pi^{2}r^{3}}{GM}$

However, IF we are considering Elliptical Orbit then it is calculated as

$T^{2} \propto a^{3}$

where 'a' is length of semi-major axis in orbit in which Planet is revolving.

Now ,Going to the question :D,

Here ,Although I am considering Distance from sun is either r or a as used above.

Then ,we get using eq. (1) ,

$\frac{T_{Uranus}^{2}}{T_{Pluto}^{2}} = \frac{20^{3}}{40^{3}} \\ \\ => \frac{T_{Uranus}}{T_{Pluto}}=\frac{1}{2\sqrt{2}}$

Hence, we get our desired ratio as

$\boxed{\frac{1}{2\sqrt{2}}} }$