The gravitational force between two bodies of masses $m_1$ and $m_2$ separated by a distance R is given by Newton's Law of Gravitation: $F = G \frac{m_1 m_2}{R^2}$, where G is the gravitational constant. In the first system: $m_1 = M$, $m_2 = M$, $R_1 = R$. The force is $F_1 = G \frac{M \times M}{R^2} = G \frac{M^2}{R^2}$. This force is given as F. So, $F = G \frac{M^2}{R^2}$. In the second system: $m_1' = 2M$, $m_2' = 2M$, $R_2 = R/2$. The force is $F_2 = G \frac{(2M) \times (2M)}{(R/2)^2}$. Calculate $F_2$: $F_2 = G \frac{4M^2}{R^2/4} = G \frac{4M^2}{R^2} \times 4 = 16 G \frac{M^2}{R^2}$. Substitute the expression for F: $F_2 = 16 F$.
Two bodies of mass M each are placed R distance apart. In another system, two bodies of mass 2M each are placed $\frac{R}{2}$ distance apart. If F be the gravitational force between the bodies in the first system, then the gravitational force between the bodies in the second system will be
Asked in UPSC NDA 2 2019 · Topic: Mechanics (Gravitation and Central Force Motion (Kepler's Laws))
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