Two satellites P and Q are moving in different circular orbits around the Earth (radius π
). The heights of P and Q from the Earth surface are \(β_P\) and \(β_Q\), respectively, where \(β_P = \frac{π
}{3}\). The accelerations of P and Q due to Earthβs gravity are \(π_P\) and \(π_Q\), respectively. If \(\frac{π_P}{π_Q} = \frac{36}{25}\), what is the value of \(β_Q?\)
\(\frac{3π }{5}\)
\(\frac{π }{6}\)
\(\frac{6π }{5}\)
\(\frac{5π }{6}\)
The Correct Option is A
Solution and Explanation
Given \(h_p=\frac{R}{3}\)
\(h_q=?\)
gravitational acceleration at height
\(g_{ht}=\frac{GM}{(R+h)^2}\)
\(\frac{g_P}{g_Q}=\frac{36}{25}\)
\(=\frac{\frac{GM}{(R+h_p)^2}}{\frac{GM}{(R+h_Q)^2}}\)
when \(h_P=\frac{R}{3}\)
on solving we get
\(h_Q=\frac{3R}{5}\)
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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.
Newtonβs Law of Gravitation
According to Newtonβs law of gravitation, βEvery particle in the universe attracts every other particle with a force whose magnitude is,
- F β (M1M2) . . . . (1)
- (F β 1/r2) . . . . (2)
On combining equations (1) and (2) we get,
F β M1M2/r2
F = G Γ [M1M2]/r2 . . . . (7)
Or, f(r) = GM1M2/r2
The dimension formula of G is [M-1L3T-2].