A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to then choose the correct option

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Q.
A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to $R^{- 3 / 2}$ then choose the correct option [2024]

1 $T^{2} \propto R^{5 / 2}$

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

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

4 $T^{2} \propto R^{3}$

Ans.
(1)

$∵$ $F \propto \frac{1}{R^{3 / 2}}$ or $F = \frac{K}{R^{3 / 2}}$

Gravitational force provides centripetal force.

$\frac{K}{R^{3 / 2}} = m ω^{2} R$

$ω^{2} = \frac{K}{m R^{5 / 2}}$

${(\frac{2 π}{T})}^{2} = \frac{K}{m R^{5 / 2}} \Rightarrow T^{2}$

$= \frac{4 π^{2} m R^{5 / 2}}{K} \Rightarrow T^{2} \propto R^{5 / 2}$

Solution

Q.
A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to $R^{- 3 / 2}$ then choose the correct option [2024]

1 $T^{2} \propto R^{5 / 2}$

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

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

4 $T^{2} \propto R^{3}$

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Solution

Q. A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to R-3/2 then choose the correct option [2024]

1 T2∝R5/2

2 T2∝R7/2

3 T2∝R3/2

4 T2∝R3

Ans. (1) ∵ F∝1R3/2 or F=KR3/2 Gravitational force provides centripetal force. KR3/2=mω2R ω2=KmR5/2 (2πT)2=KmR5/2⇒T2 =4π2mR5/2K⇒T2∝R5/2

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Q.
A light planet is revolving around a massive star in a circular orbit of radius R with a period of revolution T. If the force of attraction between planet and star is proportional to $R^{- 3 / 2}$ then choose the correct option [2024]

1 $T^{2} \propto R^{5 / 2}$

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

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

4 $T^{2} \propto R^{3}$