Problem 683 - JEE Physics

Question

As we go from the equator to the poles, the value of \(g \ldots \ldots \ldots\) (A) Remains constant (B) Decreases (C) Increases (D) Decreases upto latitude of \(45^{\circ}\)

Step-by-Step Solution

Answer

As we move from the equator to the poles, the value of gravity (\(g\)) increases. This is because the Earth's radius decreases with increasing latitude, causing an increase in the radial gravitational force, while the centrifugal force decreases. Therefore, the net effect is an increase in the gravitational force experienced by an object as it goes from the equator towards the poles.

Step 1 - Recap the equation of gravity and its components

Gravity is the force with which the Earth attracts objects towards its center. It has two components: radial gravity and centrifugal force. The radial gravity is the force that pulls objects directly towards the Earth's center, while the centrifugal force is the force that acts on objects in a rotating frame, such as Earth. The overall gravity (g) can be expressed as the difference between the radial gravity and the centrifugal force. This difference will determine how the value of gravity changes with latitude.

Step 2 - Discuss the difference in Earth's radius and gravitational force between the equator and poles

The Earth is not a perfect sphere, but rather an oblate spheroid. This means the Earth's radius is larger at the equator and smaller at the poles due to its rotation. As a result, objects on the Earth experience more radial gravity at the poles, while objects at the equator experience a combination of radial gravity and centrifugal force. To understand how the value of gravity changes with latitude, we need to consider the difference in Earth's radius and the gravitational force acting on an object at different latitudes.

Step 3 - Explain how the value of gravity changes from equator to poles

As we move from the equator towards the poles, the Earth's radius decreases, and the radial gravitational force acting on an object increases. However, the centrifugal force, which acts in the opposite direction of radial gravity, also decreases. This is because as we move from the equator to the pole, the distance from the axis of rotation decreases, reducing the effect of the centrifugal force. So, the net effect is an increase in the overall gravitational force experienced by an object when moving from the equator towards the poles. Based on this understanding, we can now determine which option is correct.

Step 4 - Select the correct option

Considering that the value of gravity increases as we move from the equator to the poles, the correct answer is: (C) Increases

Key Concepts

Oblate Spheroid Earth

The Earth might look like a perfect round ball from space. But in reality, it's an oblate spheroid. An oblate spheroid means that the Earth is slightly flattened at the poles and bulging at the equator. It's kind of like squeezing a ball from the top and bottom.

• This shape occurs because of Earth's rotation.
• The equator is bulging outward due to this spinning.
• As a result, the radius of the Earth is larger at the equator than at the poles.

This difference in shape plays a significant role in how gravity acts on objects located at different points on the Earth's surface. Let's dive deeper into how this shape affects radial gravity and more.

Radial Gravity

Radial gravity is the force that pulls objects towards the center of the Earth. Think of it as the core component of what we feel as gravity.

• It is directed from the surface of the Earth towards its center.
• Radial gravity is slightly stronger at the poles due to the smaller radius of the Earth.
• At the equator, this force is marginally less intense because of the larger radius.

The variation in Earth's radius, due to its oblate shape, means that radial gravity differs slightly depending on where you are on the planet.

Centrifugal Force

Apart from radial gravity, another player affects how gravity feels on Earth—centrifugal force. This force arises due to the Earth's rotation. Think of it like the imaginary force you feel pushing you outward when you're on a spinning merry-go-round.

• Centrifugal force is at its maximum at the equator because that's where the Earth spins the fastest.
• As you move towards the poles, this force decreases.
• This force acts in the opposite direction of radial gravity.

The combination of Earth’s rotation and its shape means centrifugal force slightly counterbalances gravity more at the equator than at the poles. This dynamic plays a crucial role in the variation of gravitational force with latitude.

Latitude Impact on Gravity

Now that we understand the players involved, let's see how latitude affects gravity. Latitude simply refers to how far north or south you are from the equator. As you move from the equator to the poles:

• Radial gravity increases because the Earth's radius decreases.
• Centrifugal force decreases due to reduced rotational speed near the poles.

What you get is a net increase in effective gravitational force as you go from the equator to the poles. This means the value of gravity increases with an increase in latitude, making option (C) from the original exercise the correct one.