Question

There is a mine of depth about \( 3.0 \,\text{km} \). Conditions prevailing in this mine as compared to those at the surface of earth are

• A. higher air pressure, lower acceleration due to gravity
• B. higher air pressure, higher acceleration due to gravity
• C. lower air pressure, higher acceleration due to gravity
• D. lower air pressure, lower acceleration due to gravity
• E. same air pressure and acceleration due to gravity

Hint:

Inside Earth, gravity decreases linearly with depth.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks about the variation of two physical quantities, air pressure and acceleration due to gravity, with depth below the Earth's surface.
Step 2: Key Formula or Approach:
1. Air Pressure: Atmospheric pressure is caused by the weight of the column of air above a certain point. As depth increases (going down from the surface), the length and weight of the air column above increase. 2. Acceleration due to Gravity (g): The acceleration due to gravity at a depth `d` below the surface of the Earth (assuming uniform density) is given by the formula: \[ g_d = g \left(1 - \frac{d}{R_E}\right) \] where `g` is the acceleration at the surface and \(R_E\) is the radius of the Earth.
Step 3: Detailed Explanation:
Analysis of Air Pressure: As one descends into a mine, the column of air overhead becomes taller and denser. The pressure at any point is the force per unit area exerted by the weight of the air above it. Since there is more air above a point inside the mine than at the surface, the air pressure inside the mine will be higher than the pressure at the surface. Analysis of Acceleration due to Gravity: According to Newton's law of gravitation, when inside a spherical body of uniform density, the gravitational force at a distance `r` from the center is only due to the mass enclosed within the sphere of radius `r`. The gravitational effect of the outer spherical shell cancels out. At a depth `d` below the surface, the distance from the center is \(r = R_E - d\). The acceleration due to gravity at this depth, \(g_d\), is given by the formula: \[ g_d = g \left(1 - \frac{d}{R_E}\right) \] Since the depth \(d = 3.0\) km is greater than zero, the factor \(\left(1 - \frac{d}{R_E}\right)\) will be less than 1. Therefore, \(g_d<g\). This means the acceleration due to gravity inside the mine is lower than at the surface. Conclusion: Combining both findings, the conditions in the mine are higher air pressure and lower acceleration due to gravity compared to the surface.
Step 4: Final Answer:
The conditions are higher air pressure and lower acceleration due to gravity.