Two planets A and B of equal mass are having their period of revolutions TA andTB such that TA= 2TB. These planets are revolving in the circular orbits of radii rA and rB respectively. Which out of the following would be the correct relationship of their orbits?
\(\text2r^{2}_A = r^{3}_B\)
\(r^{3}_{A} = 2r^{3}_B\)
\(r^{3}_A = 4r^{3}_B\)
\(T^{2}_A - T^{2}_B = \frac{π²} {GM} ( r^{3}_B - 4r^{3}_A )\)
The Correct Option is C
Solution and Explanation
The correct answer is (C) : \(r^{3}_A = 4r^{3}_B\)
\(T_A = 2T_B\)
Now
\(T^{2}_A ∝ r^{3}_A\)
\(⇒( \frac{r_A}{r_B} )^3 = (\frac{T_A}{T_B} )^2\)
\(⇒ r^{3}_A = 4r^{3}_B\)
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\[ \begin{array}{|l|l|} \hline \text{LIST-I} & \text{LIST-II} \\ \hline \text{A. Gravitational constant} & \text{I. } [LT^{-2}] \\ \hline \text{B. Gravitational potential energy} & \text{II. } [L^2T^{-2}] \\ \hline \text{C. Gravitational potential} & \text{III. } [ML^2T^{-2}] \\ \hline \text{D. Acceleration due to gravity} & \text{IV. } [M^{-1}L^3T^{-2}] \\ \hline \end{array} \]
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Concepts Used:
Gravitation
In mechanics, the universal force of attraction acting between all matter is known as Gravity, also called gravitation, . It is the weakest known force in nature.
Newton’s Law of Gravitation
According to Newton’s law of gravitation, “Every particle in the universe attracts every other particle with a force whose magnitude is,
- F ∝ (M1M2) . . . . (1)
- (F ∝ 1/r2) . . . . (2)
On combining equations (1) and (2) we get,
F ∝ M1M2/r2
F = G × [M1M2]/r2 . . . . (7)
Or, f(r) = GM1M2/r2
The dimension formula of G is [M-1L3T-2].