A cosmonaut is circling the earth in a satellite at 7 kms at a height of 630 km above the surface of earth. Calculate the centripetal force acting on the cosmonaut if his mass is 80 kg (take `R_E=6.37xx10^6m)`

Options

A. Zero
B. 560 N
C. 600 N
D. 650 N

Explanation

To solve the problem, we need to calculate the centripetal force acting on the cosmonaut in the satellite. The formula for centripetal force (Fc) is given by:

\[ F_c = \frac{m v^2}{r} \]

where:
- \( m \) is the mass of the cosmonaut,
- \( v \) is the velocity of the satellite,
- \( r \) is the radius of the circular path (distance from the center of the Earth to the satellite).

### Step 1: Identify the given values
- Mass of the cosmonaut, \( m = 80 \, \text{kg} \)
- Velocity of the satellite, \( v = 7 \, \text{km/s} = 7 \times 10^3 \, \text{m/s} \)
- Radius of the Earth, \( R_E = 6.37 \times 10^6 \, \text{m} \)
- Height of the satellite above the Earth's surface, \( h = 630 \, \text{km} = 630 \times 10^3 \, \text{m} \)

### Step 2: Calculate the total radius (r)
The total radius \( r \) from the center of the Earth to the satellite is the sum of the Earth's radius and the height of the satellite:

\[ r = R_E + h = (6.37 \times 10^6 \, \text{m}) + (630 \times 10^3 \, \text{m}) \]

Calculating this gives:

\[ r = 6.37 \times 10^6 + 0.63 \times 10^6 = 7.0 \times 10^6 \, \text{m} \]

### Step 3: Calculate the centripetal force
Now, we can substitute the values into the centripetal force formula:

\[ F_c = \frac{m v^2}{r} \]

Substituting the known values:

\[ F_c = \frac{80 \, \text{kg} \times (7 \times 10^3 \, \text{m/s})^2}{7.0 \times 10^6 \, \text{m}} \]

Calculating \( v^2 \):

\[ v^2 = (7 \times 10^3)^2 = 49 \times 10^6 \, \text{m}^2/\text{s}^2 \]

Now substituting \( v^2 \) back into the equation for \( F_c \):

\[ F_c = \frac{80 \, \text{kg} \times 49 \times 10^6 \, \text{m}^2/\text{s}^2}{7.0 \times 10^6 \, \text{m}} \]

### Step 4: Simplify the expression
This simplifies to:

\[ F_c = \frac{3920 \times 10^6 \, \text{kg m}^2/\text{s}^2}{7.0 \times 10^6 \, \text{m}} = \frac{3920}{7} \, \text{N} \]

Calculating \( \frac{3920}{7} \):

\[ F_c = 560 \, \text{N} \]

### Final Answer
The centripetal force acting on the cosmonaut is \( 560 \, \text{N} \).

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