A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceshop. What will be the number of complete revolutions made by the spaceship in 24 hours around the planet?
[Given mass of plane `=8xx10^(22)kg`
Radius of planet `=2xx10^(6)m`
Gravitational constant `G=6.67xx10^(-11)Nm^(2)//kg^(2)`]

Options

A. 9
B. 17
C. 13
D. 11

Answer

To solve the problem of finding the number of complete revolutions made by a spaceship in 24 hours around a planet, we will follow these steps:

### Step 1: Understand the Problem
The spaceship is orbiting at a height of 20 km above the surface of a planet with a radius of \(2 \times 10^6\) m. We need to calculate how many complete revolutions it makes in 24 hours.

### Step 2: Calculate the Total Radius of Orbit
The total radius \(r\) of the orbit is the sum of the radius of the planet \(R\) and the height \(h\) of the orbit:
\[ r = R + h \]
Given:
- \(R = 2 \times 10^6 \, \text{m}\)
- \(h = 20 \, \text{km} = 20 \times 10^3 \, \text{m}\)

Calculating \(r\):
\[ r = 2 \times 10^6 \, \text{m} + 20 \times 10^3 \, \text{m} = 2.02 \times 10^6 \, \text{m} \]

### Step 3: Calculate the Gravitational Force and Centripetal Force
Using the formula for gravitational force:
\[ F_g = \frac{G M m}{r^2} \]
And the centripetal force required for circular motion:
\[ F_c = \frac{m v^2}{r} \]
Setting \(F_g = F_c\):
\[ \frac{G M m}{r^2} = \frac{m v^2}{r} \]
Cancelling \(m\) from both sides and rearranging gives:
\[ v^2 = \frac{G M}{r} \]

### Step 4: Calculate the Velocity \(v\)
Taking the square root:
\[ v = \sqrt{\frac{G M}{r}} \]
Where:
- \(G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\)
- \(M = 8 \times 10^{22} \, \text{kg}\)

Substituting the values:
\[ v = \sqrt{\frac{6.67 \times 10^{-11} \times 8 \times 10^{22}}{2.02 \times 10^6}} \]

### Step 5: Calculate the Time Period \(T\)
The time period \(T\) of the orbit can be calculated using:
\[ T = \frac{2 \pi r}{v} \]

### Step 6: Calculate the Number of Revolutions in 24 Hours
The number of revolutions \(n\) in 24 hours is given by:
\[ n = \frac{T_{\text{total}}}{T} \]
Where \(T_{\text{total}} = 24 \times 3600 \, \text{s}\).

### Step 7: Substitute and Solve
1. Calculate \(v\) using the values from Step 4.
2. Calculate \(T\) using the value of \(v\) from Step 5.
3. Finally, calculate \(n\) using the total time in seconds from Step 6.

### Final Calculation
After performing all calculations, we find that the number of complete revolutions \(n\) is approximately 11.

### Conclusion
The number of complete revolutions made by the spaceship in 24 hours around the planet is **11**.

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