Question

The acceleration due to gravity at a height 1 km above the earth is the same as at a depth d below the surface of earth. Then

• A. d= \(\frac 12\) km
• B. d=1 km
• C. d= \(\frac 32\) km
• D. d=2 km

Answer 1

The Correct Option is D

To solve this question, we compare the acceleration due to gravity at a height of 1 km above the Earth's surface with that at a depth \(d\) km below the Earth's surface. The formulae for acceleration due to gravity at these positions are:

1. Acceleration due to gravity at a height \(h\) above the Earth's surface: g_h = g_0 \left(1 - \frac{2h}{R}\right) where \(g_0\) is the acceleration due to gravity on the Earth's surface, and \(R\) is the radius of the Earth.
2. Acceleration due to gravity at a depth \(d\) below the Earth's surface: g_d = g_0 \left(1 - \frac{d}{R}\right)

According to the problem, g_h = g_d. Substituting the formulae from steps 1 and 2, we get:

g_0 \left(1 - \frac{2h}{R}\right) = g_0 \left(1 - \frac{d}{R}\right)

Since \(g_0\) cancels out from both sides, the equation simplifies to:

1 - \frac{2h}{R} = 1 - \frac{d}{R}

Solving for \(d\):

\frac{d}{R} = \frac{2h}{R}

Thus:

d = 2h

Substituting \(h = 1\) km, we get:

d = 2 \times 1 = 2 \text{ km}

Therefore, the correct answer is: d = 2 km