Problem 40 - Master Resource Book in JEE Main Physics

Question

The ratio of acceleration due to gravity at a height \(h\) above the surface of the earth and at a depth \(h\) below the surface of the earth for \(h<\) radius of earth (a) is constant (b) increases linearly with \(h\) (c) decreases linearly with \(h\) (d) decreases parabolically with \(h\)

Step-by-Step Solution

Answer

(d) decreases parabolically with \( h \)

Step 1: Understanding the Problem

We are asked to find the ratio of acceleration due to gravity at a height \( h \) above the Earth's surface and at a depth \( h \) below the Earth's surface. We need to determine how this ratio changes as \( h \) changes.

Step 2: Formula for Gravity Above the Surface

The acceleration due to gravity at a height \( h \) above the Earth's surface is given by \( g_h = g \left( \frac{R}{R+h} \right)^2 \), where \( g \) is the acceleration due to gravity at the surface and \( R \) is the radius of the Earth.

Step 3: Formula for Gravity Below the Surface

The acceleration due to gravity at a depth \( h \) below the Earth's surface is given by \( g_d = g \left( 1 - \frac{h}{R} \right) \). This is derived from the gravitational force being proportional to the enclosed mass when below the surface.

Step 4: Ratio of Gravity at Height and Depth

The ratio of gravity at a height \( h \) to gravity at a depth \( h \) is \( \frac{g_h}{g_d} = \frac{\left( \frac{R}{R+h} \right)^2}{1 - \frac{h}{R}} \).

Step 5: Examine the Behavior with Respect to h

As \( h \) increases from 0, the term \( \left( \frac{R}{R+h} \right)^2 \) decreases parabolically because it is the square of a reciprocal function, while \( 1 - \frac{h}{R} \) decreases linearly.

Step 6: Conclusion Based on Observations

The ratio \( \frac{\left( \frac{R}{R+h} \right)^2}{1 - \frac{h}{R}} \) decreases parabolically with \( h \) because the numerator is a square term which decreases faster than the linear term in the denominator.

Key Concepts

Gravitational Force

Gravitational force is a fundamental concept in physics. It is the attractive force between two masses. Everybody with mass in the universe exerts a gravitational force on every other mass. The Earth, being massive, exerts a significant gravitational pull on nearby objects, including us.

The formula for gravitational force is given by Newton's Law of Universal Gravitation: \[ F = G \frac{m_1 m_2}{r^2} \] where:

• \( F \) is the gravitational force,
• \( G \) is the universal gravitational constant,
• \( m_1 \) and \( m_2 \) are the masses of the two objects,
• \( r \) is the distance between the centers of their masses.

On Earth, this concept simplifies because one of the masses is the Earth itself. This leads to the concept of acceleration due to gravity \( g \). This simplification is important when dealing with forces near Earth's surface, where gravity is approximately constant and defined as \( 9.8 \, \text{m/s}^2 \).

This simplified gravity works well for objects on or near the Earth's surface. However, changes in elevation, like moving up a mountain or going underground, affect gravitational acceleration subtly, requiring deeper mathematical analysis.

Height and Depth Model

The behavior of gravitational force changes with variations in height and depth from the Earth's surface, captured in the height and depth model. This model accounts for how gravity varies when you go up above the Earth's surface and down beneath it.

• Above the surface: As you increase your height \( h \) above the Earth's surface, the gravitational force decreases. This is because as you move away from the surface, the distance \( R + h \) from the Earth's center increases, weakening the gravitational pull. The equation \( g_h = g \left( \frac{R}{R+h} \right)^2 \) indicates that gravity decreases with the square of the height.



• Below the surface: When you go into the Earth, to a depth \( h \), the gravitational force also changes, but differently. The force inside the Earth effectively decreases linearly. This occurs because deeper, part of the Earth's mass is above you, reducing the net gravitational force downwards. The formula for this is \( g_d = g \left( 1 - \frac{h}{R} \right) \), reflecting a linear decrease relative to depth.

Therefore, understanding this model is crucial for comprehending how gravitational acceleration shifts as you move up into the air or down into the ground.

Ratio of Gravitational Acceleration

When comparing gravitational acceleration at a height and a depth \( h \), you need to consider how each varies relative to \( h \). The ratio of these accelerations provides insight into how gravity transforms with elevation or submersion.

• The acceleration ratio \( \frac{g_h}{g_d} \) is determined by comparing the formula \( g_h = g \left( \frac{R}{R+h} \right)^2 \) and \( g_d = g \left( 1 - \frac{h}{R} \right) \).
• At higher positions, the acceleration is reduced due to the square of the distance function, while below the surface, it declines linearly.

As \( h \) increases, the term \( \left( \frac{R}{R+h} \right)^2 \) decreases; this is because the square term shrinks more rapidly than the straightforward linear decrease of \( 1 - \frac{h}{R} \).

This leads to the conclusion that the ratio of acceleration due to gravity changes parabolically with \( h \). Understanding these parabolic and linear influences is key to grasping how gravitational acceleration changes based on one's position concerning Earth's surface.