A planet has density same as that of Earth and mass is twice that of Earth. If the weight of an object on Earth is "W" then the weight on the planet is:
- \(2^{\frac{2}{3}}W\)
- \(2^{\frac{1}{3}}W\)
- \(2^{\frac{4}{3}}W\)
- \(W\)
The Correct Option is B
Solution and Explanation
The correct option is(B): \(2^{\frac{1}{3}}W\)
Planet with the mass M has radius as R and Planet with mass 2M has radius as R'
\(\rho \frac{4}{3}\pi R^{3}=M\)
\(\rho \frac{4}{3}\pi R'^{3}=2M\)
\(\Rightarrow R'=2^{\frac{1}{3}}R\)
\(=2\frac{GM}{2^{\frac{2}{3}}R^{2}}=2^{\frac{1}{3}}\frac{Gm}{R^{2}}=2^{\frac{1}{2}}W\)
\(W'=2^{\frac{1}{3}}W\)
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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].