If all objects on the equator of earth feels weightless then the duration of the day will nearly become

Options

A. 6.2 hr
B. 4.4 hr
C. 2.2 hr
D. 1.41 hr

Answer

To solve the problem, we need to determine the duration of the day if all objects at the equator of the Earth feel weightless. This means that the effective acceleration due to gravity at the equator must be zero. We will follow these steps:

### Step 1: Understand the condition for weightlessness
When an object feels weightless, the net force acting on it is zero. At the equator, this condition can be expressed as:
\[ g - R_e \omega^2 = 0 \]
where:
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( R_e \) is the radius of the Earth (approximately \( 6.4 \times 10^6 \, \text{m} \)),
- \( \omega \) is the angular velocity of the Earth.

### Step 2: Rearranging the equation
From the equation above, we can rearrange it to find the angular velocity \( \omega \):
\[ g = R_e \omega^2 \]
\[ \omega = \sqrt{\frac{g}{R_e}} \]

### Step 3: Calculate the current angular velocity
The current angular velocity \( \omega_0 \) of the Earth can be calculated using the formula:
\[ \omega_0 = \frac{2\pi}{T} \]
where \( T \) is the period of rotation (the duration of a day), which is approximately \( 86400 \, \text{s} \).

### Step 4: Find the ratio of the new angular velocity to the current angular velocity
We can find the ratio of the new angular velocity \( \omega \) to the current angular velocity \( \omega_0 \):
\[ \frac{\omega}{\omega_0} = \frac{\sqrt{\frac{g}{R_e}}}{\frac{2\pi}{T}} \]

### Step 5: Substitute the values
Substituting the known values:
- \( g \approx 9.8 \, \text{m/s}^2 \)
- \( R_e \approx 6.4 \times 10^6 \, \text{m} \)
- \( T = 86400 \, \text{s} \)

Calculating \( \omega \):
\[ \omega = \sqrt{\frac{9.8}{6.4 \times 10^6}} \]

### Step 6: Calculate the new duration of the day
Now, we need to find the new duration of the day \( T' \) when the angular velocity is \( \omega \):
\[ T' = \frac{2\pi}{\omega} \]

### Step 7: Calculate the final result
Using the ratio derived earlier, we can express the new duration of the day in terms of the current duration:
\[ T' = \frac{T}{17} \]
\[ T' = \frac{86400}{17} \approx 5082.35 \, \text{s} \approx 1.41 \, \text{hours} \]

Thus, the new duration of the day will be approximately **1.41 hours**.

### Final Answer
The duration of the day will nearly become **1.41 hours**.