Two satellites of masses m and 3m revolve around the earth in circular orbits of radii r & 3r respectively. The ratio of orbital speeds of the satellites respectively is
Show Hint
The orbital speed of a satellite depends only on the mass of the central body (in this case, Earth) and the orbital radius. The mass of the satellite itself is irrele vant. Use the formula for orbital speed to find the ratio of speeds
- √3 :1
- 1:√3
- √2 :1
- 1:2
The Correct Option is A
Solution and Explanation
The velocity of a satellite in orbit is given by:
\( v = \sqrt{\frac{GM}{r}} \)
Where:
- \( G \): Gravitational constant
- \( M \): Mass of the Earth
- \( r \): Radius of the orbit
Step 1: Proportionality Relation:
Rearranging the formula, we see that velocity is inversely proportional to the square root of the radius:
\( v \propto \frac{1}{\sqrt{r}} \)
Step 2: Comparing Velocities at Two Radii:
For two different radii, \( r_1 \) and \( r_2 \):
\( \frac{v_1}{v_2} = \sqrt{\frac{r_2}{r_1}} \)
Step 3: Substituting Values:
If \( r_2 = 3r_1 \):
\( \frac{v_1}{v_2} = \sqrt{\frac{r_2}{r_1}} = \sqrt{3} \)
Conclusion:
The velocity at the smaller radius (\( v_1 \)) is \( \sqrt{3} \) times the velocity at the larger radius (\( v_2 \)).
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