Correct Answer - Option 4 : 11
Concept:
Time taken to complete one complete revolution:
We know that V = 2πr/T is the linear velocity of a particle undergoing circular motion. 2πr is the total distance covered in one full revolution and T is the time taken for one full revolution.
Then, the time taken to complete revolution is given by the formula:
T = 2πr/v
Velocity of the object in circular orbit:
All bounded orbits where the gravity of a central body dominates are elliptical in nature. A special case of this is the circular orbit, which is an ellipse of zero eccentricity. The formula for the velocity of a body in a circular orbit (orbital speed) at distance r from the centre of gravity of mass M is \({\rm{v}} = \sqrt {\frac{{{\rm{GM}}}}{{\rm{r}}}} \)
Calculation:
Given,
Mass of a planet = 8 × 1022 kg
Radius of the planet = 2 × 106 m
Gravitational constant, G = 6.67 × 10-11 Nm2/kg2
The time taken to complete one complete revolution is given by the formula:
\({\rm{T}} = \frac{{2{\rm{\pi r}}}}{{\rm{v}}}\)
Where, the velocity of the object in circular orbit is given by the formula:
\({\rm{v}} = \sqrt {\frac{{{\rm{GM}}}}{{\rm{r}}}} \)
On substituting, the velocity in time,
\(\Rightarrow {\rm{T}} = \frac{{2{\rm{\pi r}}}}{{\sqrt {\frac{{{\rm{GM}}}}{{\rm{r}}}} }}\)
\(\Rightarrow {\rm{T}} = 2{\rm{\pi r}}\left( {\sqrt {\frac{{\rm{r}}}{{{\rm{GM}}}}} } \right)\)
\(\Rightarrow {\rm{T}} = 2{\rm{\pi }}\sqrt {\frac{{{{\rm{r}}^3}}}{{{\rm{GM}}}}} \)
Where,
G = Gravitation constant
M = Mass of the planet
r = Radius of the planet
Now, substituting the values
\( \Rightarrow {\rm{T}} = 2{\rm{\pi }}\sqrt {\frac{{{{\left( {2 \times {{10}^6}} \right)}^3}}}{{\left( {6.67 \times {{10}^{ - 11}}} \right)\left( {8 \times {{10}^{22}}} \right)}}} \)
⇒ T = 2π × 1224.4
∴ T = 7693.37
The number of complete revolutions in 24 hours around the planet is given by the formula:
\({\rm{n}} = \frac{{24 \times 60 \times 60}}{{\rm{T}}}\)
\(\Rightarrow {\rm{n}} = \frac{{24 \times 60 \times 60}}{{7693.37}}\)
∴ n = 11.2 ≈ 11
