3. Assuming the earth is made of matter of uniform density and has radius 8000 km,
the acceleration due to gravity in a 2000 km deep mine would be:
a. 2.45 m/s2
b. 7.36 m/s2
c. 9.81 m/s2
d. The answer cannot be determined from this information

Acceleration Due to Gravity in a 2000 km Deep Mine: CSIR NET Solved

The acceleration due to gravity in a 2000 km deep mine on an Earth of uniform density and 8000 km radius is 7.36 m/s².

Key Concept

For a uniform density sphere, gravity at the surface follows g = ⁴/₃πGρR = ³/₄πGρR, where ρ is density, G is the gravitational constant, and R is radius.

Inside at distance r from center, only mass within r contributes, yielding g(r) = ⁴/₃πGρr = ^{g r}/_R or g(r) = ³/₄πGρr = g ^r/_R.

Step-by-Step Solution

• Earth radius R = 8000 km, mine depth d = 2000 km, so r = R – d = 6000 km.
• Ratio ^r/_R = 6000/8000 = 0.75.
• Thus, g_mine = g_surface × 0.75.
• Standard g_surface ≈ 9.81 m/s² gives 0.75 × 9.81 = 7.36 m/s² exactly matching option b.

Option Analysis

• a. 2.45 m/s²: Too low; matches zero-gravity center or incorrect r/R = 0.25.
• b. 7.36 m/s²: Correct, as g(r) = g × 0.75.
• c. 9.81 m/s²: Surface value; ignores reduced enclosed mass inside Earth.
• d. Cannot be determined: Incorrect; uniform density and dimensions suffice for ratio calculation without G or ρ.