Question

At what distance above and below the surface of the earth a body will have the same weight (take radius of earth as R)?

• A. \(\frac{R}{2}\)
• B. \((\sqrt5-1)\frac{R}{2}\)
• C. \((\sqrt3-1)\frac{R}{2}\)
• D. \((\sqrt5-1)R\)

Answer 1

The Correct Option is B

Step 1: Define Gravitational Acceleration Above and Below: - Above the earth’s surface at height h, gravitational acceleration g_p is:

\( g_{p} = \frac{gR^2}{(R + h)^2} \)

- Below the earth’s surface at depth h, gravitational acceleration g_q is:

\( g_{q} = g \left(1 - \frac{h}{R}\right) \)

Step 2: Set g_p = g_q:

\( \frac{gR^2}{(R + h)^2} = g \left(1 - \frac{h}{R}\right) \)

Step 3: Simplify the Equation:

\( \frac{1}{\left(1 + \frac{h}{R}\right)^2} = 1 - \frac{h}{R} \)

\( \left(1 - \frac{h}{R}\right) \left(1 + \frac{h}{R}\right) = 1 \)

Step 4: Let \( \frac{h}{R} = x \):

\( (1 - x)(1 + x) = 1 \)

\( 1 - x^2 = 1 \)

\( x = \frac{\sqrt{5} - 1}{2} \)

Step 5: Calculate h:

\( h = \frac{R}{2}(\sqrt{5} - 1) \)

So, the correct answer is: \( h = \frac{R}{2}(\sqrt{5} - 1) \)