How far below the Earth's level will the value of gravitational acceleration be half of the gravitational acceleration on the Earth's surface ? (Radius of Earth 6400 km)

Question

Accepted Answer

To determine how far below the Earth's surface the gravitational acceleration is half of its value on the surface, we need to understand how gravitational acceleration changes with depth inside the Earth.

1. **Explanation for Option 2 (3200 km):**

The gravitational acceleration $ g $ at a distance $ r $ from the center of the Earth is given by the formula:
   $$
   g = \frac{G \cdot M(r)}{r^2}
   $$
   where $ G $ is the gravitational constant, and $ M(r) $ is the mass enclosed within radius $ r $.

Inside the Earth, assuming uniform density, the gravitational acceleration decreases linearly with depth. At the surface, the gravitational acceleration is $ g_0 $. If we want the gravitational acceleration to be half of $ g_0 $, we set:
   $$
   \frac{g_0}{2} = g_0 \cdot \left(1 - \frac{d}{R}\right)
   $$
   where $ d $ is the depth below the surface, and $ R $ is the Earth's radius (6400 km).

Solving for $ d $:
   $$
   \frac{1}{2} = 1 - \frac{d}{6400}
   $$
   $$
   \frac{d}{6400} = \frac{1}{2}
   $$
   $$
   d = 3200 \text{ km}
   $$

Therefore, the correct answer is Option 2, which is 3200 km.

2. **Explanation for why other options are incorrect:**

- **Option 1 (6400 km):** This option suggests that the gravitational acceleration would be half at the center of the Earth. However, at the center of the Earth, the gravitational acceleration is actually zero because the mass is symmetrically distributed around the center, canceling out the gravitational forces.

- **Option 3 (2650 km):** This option suggests a depth that is less than 3200 km. At 2650 km, the gravitational acceleration would be more than half of its surface value because the decrease in gravitational acceleration is linear with depth. Therefore, this depth is insufficient to reduce the gravitational acceleration to half.

- **Option 4 (1600 km):** This option suggests a depth that is even less than 3200 km. At 1600 km, the gravitational acceleration would be significantly more than half of its surface value. Thus, this depth is also insufficient to achieve the desired reduction in gravitational acceleration.

In conclusion, Option 2 (3200 km) is the correct answer because it accurately represents the depth at which the gravitational acceleration is half of its value on the Earth's surface, based on the linear decrease of gravitational acceleration with depth inside the Earth.