Question

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)

• A. \( g' = g \left( 1 - \frac{h^2}{2R^2} \right) \)
• B. \( g' = g \left( 1 - \frac{2h}{R} \right) \)
• C. \( g' = g \left( 1 - \frac{h}{2R} \right) \)
• D. \( g' = g \left( 1 - \frac{2h^2}{R^2} \right) \)

Answer 1

The Correct Option is B

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: \[ \boxed{g' = g \left( 1 - \frac{2h}{R} \right)}. \]