Question:
Two planets of radii $R_1$ and $R_2$ have the same density. The ratio of the acceleration due to gravity on these planets is
Two planets of radii $R_1$ and $R_2$ have the same density. The ratio of the acceleration due to gravity on these planets is
Updated On: May 12, 2024
- $\frac{R_1}{R_2}$
- $\frac{R_2}{R_1}$
- $\left(\frac{R_2}{R_1}\right)^2$
- $\left(\frac{R_1}{R_2}\right)^2$
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The Correct Option is A
Solution and Explanation
We know acceleration due to gravity of a planet is given by,
$g = \frac{GM}{R^{2}} = \frac{G\rho \frac{4}{3} \pi R^{3}}{R^{2}}$
or, $g = \frac{4}{3}\pi\rho GR $ or , $g \propto R$
$\therefore\:\:\: \frac{g_{1}}{g_{2}} = \frac{R_{1}}{R_{2}}$
$g = \frac{GM}{R^{2}} = \frac{G\rho \frac{4}{3} \pi R^{3}}{R^{2}}$
or, $g = \frac{4}{3}\pi\rho GR $ or , $g \propto R$
$\therefore\:\:\: \frac{g_{1}}{g_{2}} = \frac{R_{1}}{R_{2}}$
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Concepts Used:
Newtons Law of Gravitation
Gravitational Force
Gravitational force is a central force that depends only on the position of the test mass from the source mass and always acts along the line joining the centers of the two masses.
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,
- Directly proportional to the product of their masses i.e. F ∝ (M1M2) . . . . (1)
- Inversely proportional to the square of the distance between their center i.e. (F ∝ 1/r2) . . . . (2)
By combining equations (1) and (2) we get,
F ∝ M1M2/r2
F = G × [M1M2]/r2 . . . . (7)
Or, f(r) = GM1M2/r2 [f(r)is a variable, Non-contact, and conservative force]