Problem 58 - Master Resource Book in JEE Main Physics

Question

The effect of rotation of the earth on the value of acceleration due to gravity is (a) \(g\) is maximum at the equator and minimum at the poles (b) \(g\) is minimum at the equator and maximum at the poles (c) \(g\) is maximum at both places (d) \(g\) is minimum at both places

Step-by-Step Solution

Answer

(b) \(g\) is minimum at the equator and maximum at the poles.

Step 1: Understand Gravitational Acceleration

Gravitational acceleration, represented as \(g\), is affected by the Earth's rotation. The centrifugal force due to the Earth's rotation influences the effective value of \(g\) at any point on the Earth's surface.

Step 2: Analyze Centrifugal Force Effect

At the equator, the centrifugal force is maximum because the distance to the axis of rotation is greatest, reducing the effective gravity \(g\). At the poles, this centrifugal force is zero because the poles lie on the axis of rotation, so \(g\) is only influenced by gravitational force.

Step 3: Compare Gravity at Equator and Poles

From the centrifugal force analysis, it's clear that \(g\) is lower at the equator than at the poles because the effective \(g\) at the equator is reduced by the centrifugal force.

Step 4: Conclusion

Combine the findings to conclude that \(g\) is minimum at the equator due to the rotational effect and maximum at the poles where the gravitational force is unopposed by centrifugal force.

Key Concepts

Centrifugal Force

The concept of centrifugal force is crucial when exploring how Earth's rotation influences gravity. Imagine you're on a merry-go-round. You feel pushed outward, away from the center. This feeling is caused by centrifugal force. On Earth, a similar effect happens due to its rotation.
Earth spins around an axis, and this spin creates centrifugal force that acts outward from the axis of rotation. The strength of this force varies depending on your position on Earth. The closer you are to the equator, the stronger the centrifugal force because you are further from the axis. Conversely, at the poles, this force essentially disappears because you're directly on the axis.

• Equator: Maximum centrifugal force due to greatest distance from the axis.
• Poles: Zero centrifugal force because the distance from the axis is zero.

Understanding centrifugal force helps in grasping why the effective gravity differs from place to place on our spinning planet.

Gravitational Acceleration

Gravitational acceleration, often denoted as \( g \), is the acceleration of an object due to Earth's gravity. This value isn't constant everywhere on earth because of factors like Earth's rotation and its shape. Rotation, in particular, has an intriguing effect because it introduces the centrifugal force mentioned earlier.
The nominal value of \( g \) is approximately 9.81 m/s². However, because of the centrifugal force acting opposite to gravity, especially at lower latitudes, the effective gravitational acceleration is slightly less than this nominal value.

• Effective Gravity: The actual magnitude of gravitational pull experienced after considering centrifugal effects.
• Variation: Effective \( g \) is less than 9.81 m/s² at the equator, slightly rising as one moves towards the poles.

The measured value of \( g \) accounts for both gravitational pull and centrifugal force, giving a slightly varied reading depending on location.

Gravity at Equator and Poles

The difference in gravity's strength at the equator versus the poles is fascinating. The defining reason for this difference lies in the centrifugal force affecting the gravitational pull. At the equator, you encounter the maximum effect of centrifugal force because of the Earth's equatorial bulge and distance from the rotation axis. This force acts against gravity, effectively reducing your weight and the apparent gravitational force.
In contrast, when you're standing at one of the poles, you're on the axis of Earth's rotation, which means centrifugal force doesn't play any role in opposing gravity there. Consequently, the only force you experience is gravitational, and thus gravity is stronger at the poles.

• At Equator: Lower gravitational force due to maximum opposing centrifugal force.
• At Poles: Higher gravitational force as there's no centrifugal force contribution.

This variation helps us understand why objects can weigh slightly less at the equator compared to the poles, highlighting the impact of Earth's rotation on gravitational forces.