Which of the following expressions give the value of acceleration due to gravity (g') at the altitude h above the surface of Earth.
(R = radius of Earth, g = acceleration due to gravity at surface of Earth)
(R = radius of Earth, g = acceleration due to gravity at surface of Earth)
- \( g' = g \left( 1 - \frac{h^2}{2R^2} \right) \)
- \( g' = g \left( 1 - \frac{2h}{R} \right) \)
- \( g' = g \left( 1 - \frac{h}{2R} \right) \)
- \( g' = g \left( 1 - \frac{2h^2}{R^2} \right) \)
The Correct Option is B
Solution and Explanation
The acceleration due to gravity at a height \( h \) above the Earth's surface is: \[ g' = \frac{GM}{(R + h)^2}. \] Using the binomial expansion for \( (1 + \frac{h}{R})^2 \), neglecting higher-order terms: \[ \frac{1}{(1 + \frac{h}{R})^2} \approx 1 - \frac{2h}{R}. \] Substituting this approximation: \[ g' = \frac{GM}{R^2} \left( 1 - \frac{2h}{R} \right). \] Since \( g = \frac{GM}{R^2} \), we can write: \[ g' = g \left( 1 - \frac{2h}{R} \right). \]
Final Answer: The acceleration due to gravity is: \[ \boxed{g' = g \left( 1 - \frac{2h}{R} \right)}. \]
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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].