Question:
If a planet has twice the mass of earth and three times the radius $(R)$ of earth, then the escape velocity of the planet is ($\upsilon_e$ = escape velocity of earth)
If a planet has twice the mass of earth and three times the radius $(R)$ of earth, then the escape velocity of the planet is ($\upsilon_e$ = escape velocity of earth)
Updated On: Jun 7, 2022
- $\sqrt{ \frac{1}{2}} \upsilon_e$
- $\sqrt{ \frac{2}{3}} \upsilon_e$
- $\sqrt{2} \upsilon_e$
- none of these
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The Correct Option is B
Solution and Explanation
Escape velocity,$\upsilon_e = \sqrt{2gR}$
$ = \sqrt{\frac{2GM}{R^2}} R$ $\left( \because \:\: g = \frac{GM}{R^2} \right)$
$\therefore \:\: \upsilon_e \propto \sqrt{\frac{M}{R}}$ ...(i)
Given, $M_p = 2M$ and $R_p = 3R$
$\therefore \:\:\: (\upsilon_e)_p = \sqrt{\frac{2}{3}} \upsilon_e$ (Using (i))
$ = \sqrt{\frac{2GM}{R^2}} R$ $\left( \because \:\: g = \frac{GM}{R^2} \right)$
$\therefore \:\: \upsilon_e \propto \sqrt{\frac{M}{R}}$ ...(i)
Given, $M_p = 2M$ and $R_p = 3R$
$\therefore \:\:\: (\upsilon_e)_p = \sqrt{\frac{2}{3}} \upsilon_e$ (Using (i))
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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].