The angular speed of earth's rotation about its own axis is `omega`. When its angular speed is increased to n time its original angular speed, the acceleration due to gravity at the equator becomes zero. What is the value of n?
[R is the equatorial radius of the earth]

Options

A. Increases to n times
B. Decreases to times
C. Decreases to times
D. (1 / `omega`) `sqrt` (`g/R`)

Answer

To solve the problem, we need to analyze the forces acting on a mass at the equator of the Earth when its angular speed is increased.

### Step-by-Step Solution:

1. **Understanding the Forces**:
At the equator, the gravitational force acting on a mass \( m \) is \( mg \) (where \( g \) is the acceleration due to gravity). When the Earth rotates, there is a centrifugal force acting outward, which can be expressed as \( F_c = m \omega^2 R \), where \( R \) is the equatorial radius of the Earth and \( \omega \) is the angular speed.

2. **Setting Up the Equation**:
Initially, the gravitational force is balanced by the centrifugal force:
\[ mg = m \omega^2 R \]
Here, we can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ g = \omega^2 R \]

3. **Increasing Angular Speed**:
When the angular speed is increased to \( n \) times its original speed, the new angular speed becomes \( \omega' = n\omega \). The new centrifugal force acting on the mass becomes:
\[ F_c' = m (n\omega)^2 R = m n^2 \omega^2 R \]

4. **Condition for Zero Effective Gravity**:
The problem states that at this new angular speed, the effective acceleration due to gravity at the equator becomes zero. This means that the gravitational force is completely balanced by the centrifugal force:
\[ mg = m n^2 \omega^2 R \]
Again, we can cancel \( m \):
\[ g = n^2 \omega^2 R \]

5. **Equating the Two Expressions for \( g \)**:
From the first equation, we have \( g = \omega^2 R \). Setting the two expressions for \( g \) equal to each other gives:
\[ \omega^2 R = n^2 \omega^2 R \]

6. **Solving for \( n \)**:
We can divide both sides by \( \omega^2 R \) (assuming \( \omega^2 R \neq 0 \)):
\[ 1 = n^2 \]
Taking the square root of both sides:
\[ n = 1 \]

### Final Answer:
The value of \( n \) is \( 1 \).