Problem 50 - Master Resource Book in JEE Main Physics

Question

For a body lying on the equator to appear weightless, what should be the angular speed of the earth? (Take \(g=10 \mathrm{~ms}^{-2}\); radius of earth \(=6400 \mathrm{~km}\) ) (a) \(0.125 \mathrm{rads}^{-1}\) (b) \(1.25 \mathrm{rads}^{-1}\) (c) \(1.25 \times 10^{-3} \mathrm{rads}^{-1}\) (d) \(1.25 \times 10^{-2} \mathrm{rads}^{-1}\)

Step-by-Step Solution

Answer

The angular speed should be \(1.25 \times 10^{-3} \mathrm{rads}^{-1}\) (option c).

Step 1: Understanding the Problem

For a body lying on the equator to appear weightless, the centrifugal force due to Earth's rotation must equal the gravitational force acting on the body.

Step 2: Formula for Centrifugal Force

Use the formula for centrifugal force, which is \[ F_c = m \cdot r \cdot \omega^2 \]where \( m \) is mass, \( r \) is radius of Earth, and \( \omega \) is angular speed.

Step 3: Formula for Gravitational Force

The gravitational force can be calculated using \[ F_g = m \cdot g \]where \( m \) is mass and \( g = 10 \, \mathrm{m/s^2} \) is the acceleration due to gravity.

Step 4: Setting Forces Equal

For the body to appear weightless, equate the centrifugal and gravitational forces: \[ m \cdot r \cdot \omega^2 = m \cdot g \]Cancelling out \( m \) (assuming mass is non-zero) from both sides, \[ r \cdot \omega^2 = g \].

Step 5: Solve for Angular Speed

To find \( \omega \), rearrange and solve the equation \[ \omega^2 = \frac{g}{r} \]Substituting \( r = 6400 \, \mathrm{km} = 6400000 \, \mathrm{m} \): \[ \omega^2 = \frac{10}{6400000} \]

Step 6: Calculating Angular Speed

Simplify and take the square root: \[ \omega^2 = 1.5625 \times 10^{-6} \cdot \mathrm{sec}^{-2} \]\[ \omega = \sqrt{1.5625 \times 10^{-6}} \approx 1.25 \times 10^{-3} \mathrm{rad/s} \].

Step 7: Select the Correct Answer

The calculated angular speed \( 1.25 \times 10^{-3} \mathrm{rads}^{-1} \) matches option (c).

Key Concepts

Centrifugal Force

Centrifugal force is an apparent force that acts outward on a body moving around a center, arising from the body's inertia. Imagine you are on a merry-go-round. As it spins faster, you feel as if you are being pushed outward. This is due to the centrifugal force.
When talking about the Earth, this concept applies to objects at different latitudes. Because the Earth rotates, objects at the equator experience more centrifugal force than those nearer to the poles. This is due to the increased rotational speed at the equator.

• The centrifugal force formula is given by: \[ F_c = m \cdot r \cdot \omega^2 \ \text{where } m \text{ is mass, } r \text{ is radius, and } \omega \text{ is angular speed.} \]

This force is significant enough at the equator that it can partially offset the gravitational force. In the context of this exercise, the goal is to determine the specific angular speed needed to make this force equal the Earth's gravitational pull, effectively making the object "weightless."

Gravitational Force

Gravitational force is a natural phenomenon by which all things with mass or energy are brought toward one another. On Earth, it gives weight to physical objects and causes them to fall toward the ground when dropped.
The strength of this force can be calculated with the formula:

• \( F_g = m \cdot g \) \[ \text{where } m \text{ is mass and } g = 10 \, \mathrm{m/s^2} \text{ is the acceleration due to gravity.} \]

In the context of the exercise, to make a body appear weightless at the equator, the centrifugal force due to Earth's spin must match this gravitational force. Weightlessness can be thought of as a condition in which the net gravitational force is balanced or nullified by another force, such as the centrifugal force created by rotation.

Equator

The equator is an imaginary line around the middle of the Earth. It is equidistant from the North and South Poles and serves as the latitude 0° line.
Due to the Earth's rotation, objects located along the equator travel the farthest and fastest compared to objects at other latitudes. This means they experience the strongest centrifugal force.

• The radius of the Earth is about 6400 km or 6400000 m at the equator.

Therefore, to make an object at the equator appear weightless, the centrifugal force (which increases with more angular speed) must balance the gravitational pull. This is why questions about weightlessness often specify locations at the equator, where Earth's rotational effects are maximized. Understanding the interplay of these forces at the equator is crucial for solving problems related to angular speed and weightlessness.