Problem 29 - Master Resource Book in JEE Main Physics

Question

The height at which the acceleration due to gravity decreases by \(36 \%\) of its value on the surface of the earth. (The radius of the earth is \(R\) ). (a) \(\frac{R}{6}\) (b) \(\frac{R}{4}\) (c) \(\frac{R}{2}\) (d) \(\frac{2}{3} R\)

Step-by-Step Solution

Answer

d

Step 1: Understanding Gravity’s Variation with Height

The acceleration due to gravity (abla gabla gdabla g = \frac{GM}{R^2} \frac{GM}{R^2} \frac{GM}{R^2} \frac{GM}{R^2}) is inversely proportional to the square of the distance from the center of the Earth, meaning it decreases with increasing height. It is related to the height above the surface by \( g_h = g \left( \frac{R}{R + h} \right)^2 \) where \( g_h \) is the gravity at height \( h \).

Key Concepts

Gravitational Force

Gravitational force is a fundamental force of nature that attracts two masses towards each other. This force happens because any object with mass exerts a force on other objects with mass.
The Earth, being massive, exerts a significant gravitational force that pulls objects towards its center. The formula for gravitational force is given by Newton's law of universal gravitation:

• Formula: \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \)
• Where \( F \) is the gravitational force, \( G \) is the universal gravitational constant, \( m_1 \) and \( m_2 \) are the masses involved, and \( r \) is the distance between the centers of the two masses.

By understanding this, it's clear that gravitational force decreases when the distance between two masses increases. This principle helps explain why as you move farther from the Earth's surface, the gravitational pull weakens.
This is an important concept to grasp when studying how gravity changes with altitude.

Variation of Gravity with Altitude

When you increase your altitude, say by climbing a mountain or flying in an airplane, the gravitational force exerted by the Earth decreases. Simply put, the further you are from the Earth's center, the less gravitational pull there is on you.
Gravity on the Earth's surface is strongest at sea level. As you ascend, the strength of gravity diminishes. The rate at which gravity decreases with altitude can be estimated using the formula:

• \( g_h = g \left( \frac{R}{R + h} \right)^2 \)
• Where \( g_h \) is the gravity at height \( h \), \( g \) is the standard gravity at the Earth's surface, and \( R \) is the Earth's radius.

This formula shows that gravity decreases significantly only at large heights compared to the Earth’s radius.
For everyday altitudes like the height of Mount Everest, the change in gravitational acceleration is relatively small.

Inverse Square Law

The inverse square law is a pivotal principle in physics that describes how a physical quantity decreases with the square of the distance from the source. This concept heavily influences how gravity operates in our universe.
For gravity, this law states that the force decreases proportionally to the square of the distance from the Earth's center. When we talk about the inverse square law in terms of gravity:

• As the distance \( r \) increases, gravitational force \( F \) decreases as \( \frac{1}{r^2} \).
• For example, doubling the distance from the Earth's center will cause the gravitational force to decrease by a factor of four (\(2^2\)).

This is why gravity becomes weaker at higher altitudes and distances.
Understanding the inverse square law helps in computing gravity's weakening effect as one moves away from the Earth's surface, providing significant insights into how our world works.