An object weighs `10 N` at north pole of Earth. In a geostationary satellite distance `7 R` from centre of Earth (of radius `R`), what will be its (a) true weight (b) apparent weight?

Options

A. 0 N, 0N
B. 0.2N,0
C. 0.2N.9.8N
D. 0.2N,0.2N

Explanation

To solve the problem, we need to determine both the true weight and the apparent weight of an object in a geostationary satellite located at a distance of \(7R\) from the center of the Earth, where \(R\) is the radius of the Earth. The object weighs \(10 N\) at the North Pole.

### Step 1: Understand the Given Information
- Weight of the object at the North Pole, \( W = 10 \, \text{N} \)
- Distance from the center of the Earth to the satellite, \( d = 7R \)
- Height of the satellite above the Earth's surface, \( h = d - R = 7R - R = 6R \)

### Step 2: Calculate the Mass of the Object
The weight of the object at the North Pole is given by the formula:
\[ W = mg \]
where \( m \) is the mass and \( g \) is the acceleration due to gravity at the North Pole (approximately \(9.8 \, \text{m/s}^2\)).

To find the mass \( m \):
\[ m = \frac{W}{g} = \frac{10 \, \text{N}}{9.8 \, \text{m/s}^2} \approx 1.02 \, \text{kg} \]

### Step 3: Calculate the True Weight in the Satellite
The acceleration due to gravity at a distance \( d \) from the center of the Earth is given by:
\[ g' = \frac{gR^2}{d^2} \]
Substituting \( d = 7R \):
\[ g' = \frac{gR^2}{(7R)^2} = \frac{gR^2}{49R^2} = \frac{g}{49} \]
Now substituting \( g \approx 9.8 \, \text{m/s}^2 \):
\[ g' \approx \frac{9.8}{49} \approx 0.2 \, \text{m/s}^2 \]
Now, the true weight \( W_s \) in the satellite is:
\[ W_s = mg' = m \cdot \frac{g}{49} = 1.02 \cdot 0.2 \approx 0.2 \, \text{N} \]

### Step 4: Calculate the Apparent Weight in the Satellite
In a geostationary satellite, the satellite is in free fall, and the apparent weight \( W_a \) is given by:
\[ W_a = m(g' - a) \]
where \( a \) is the acceleration of the satellite. Since the satellite is in free fall, \( a = g' \). Therefore:
\[ W_a = m(g' - g') = m \cdot 0 = 0 \, \text{N} \]

### Final Results
- (a) True Weight: \( W_s \approx 0.2 \, \text{N} \)
- (b) Apparent Weight: \( W_a = 0 \, \text{N} \)