Two satellites of earth, $ S_1 $ and $ S_2 $ , are moving in the same orbit. The mass of $ S_1 $ is four times the mass of $ S_2 $ . Which one of the following statements is true ?
- The time period of $ S_1 $ is four times that of $ S_2 $
- The potential energies of earth and satellite in the two cases are equal
- $ S_1 $ and $ S_2 $ are moving with the same speed
- The kinetic energies of the two satellites are equal
The Correct Option is C
Solution and Explanation
\(K.E. =\frac{ GMm }{2r}\) \(\Rightarrow\) Kinetic energies are unequal.
\(T =\frac{2 \pi r ^{\frac 32}}{\sqrt{ GM }}\) \(\Rightarrow\) Time period are equal.
P.E. \(=-\frac{\text {GMm}}{r}\) \(\Rightarrow\) Potential energies are unequal.
\(v =\sqrt{\frac{ GM }{ r }}\) \(\Rightarrow\) Orbital speeds are equal.
So, the cortrect option is (C): \(S_1\) and \(S_2\) are moving with the same speed.
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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].