First slide

Motion of planets and satellites

Question

The period of revolution of a planet around the sun in circular orbit is same as that of period of similar planet revolving around a star of twice the radius of first orbit and ‘M’ is the mass of the sun and mass of star is

Easy

A

2M
B

4M
C

8M
D

16M

Solution

$G r a v i t a t i o n a l f o r c e p r o v i d e c e n t r i p e t a l f o r c e, m r ω^{2} = \frac{G M m}{r^{2}} \Rightarrow ω^{2} = \frac{G M}{r^{3}} \Rightarrow ω = \sqrt{\frac{G M}{r^{3}}} \Rightarrow \frac{2 π}{T} = \sqrt{\frac{G M}{r^{3}}} \Rightarrow T = 2 π \sqrt{\frac{r^{3}}{G M}}$ $T \propto \sqrt{\frac{r^{3}}{M}} \Rightarrow \frac{T_{2}}{T_{1}} = \sqrt{{(\frac{r_{2}}{r_{1}})}^{3} \frac{M_{1}}{M_{2}}} \Rightarrow 1 = \sqrt{{(\frac{2 r_{1}}{r_{1}})}^{3} \frac{M_{1}}{M_{2}}} \Rightarrow 1 = {(2)}^{3} \frac{M_{1}}{M_{2}} \Rightarrow M_{2} = 8 M_{1}$

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