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A satellite is launched into a circular orbit of radius R while a second satellite is launched into an orbit of radius 1.02R. What is the percentage change in the time periods of the two satellites?
- a)0.7
- b)1
- c)1.5
- d)3
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
A satellite is launched into a circular orbit of radius R while a seco...
Given:
- Radius of the first satellite's orbit = R
- Radius of the second satellite's orbit = 1.02R
To find:
- Percentage change in the time periods of the two satellites
Formula:
- The time period of a satellite in a circular orbit is given by T = 2π√(r³/GM), where r is the radius of the orbit, G is the gravitational constant, and M is the mass of the central body.
Calculation:
For the first satellite:
- Radius of the orbit, r₁ = R
- Time period, T₁ = 2π√(R³/GM) ------------- (1)
For the second satellite:
- Radius of the orbit, r₂ = 1.02R
- Time period, T₂ = 2π√((1.02R)³/GM) ------------ (2)
Percentage change in the time periods:
- Percentage change = (|T₁ - T₂|/T₁) * 100%
Substituting the values from equations (1) and (2):
- Percentage change = (|2π√(R³/GM) - 2π√((1.02R)³/GM)| / (2π√(R³/GM))) * 100%
- Simplifying the expression:
= (|√(R³) - √((1.02R)³)| / √(R³)) * 100%
= (|R - 1.02R| / R) * 100%
= (|-0.02R| / R) * 100%
= (0.02R / R) * 100%
= 0.02 * 100%
= 2%
Therefore, the percentage change in the time periods of the two satellites is 2%, which is closest to option D (3%).
- Radius of the first satellite's orbit = R
- Radius of the second satellite's orbit = 1.02R
To find:
- Percentage change in the time periods of the two satellites
Formula:
- The time period of a satellite in a circular orbit is given by T = 2π√(r³/GM), where r is the radius of the orbit, G is the gravitational constant, and M is the mass of the central body.
Calculation:
For the first satellite:
- Radius of the orbit, r₁ = R
- Time period, T₁ = 2π√(R³/GM) ------------- (1)
For the second satellite:
- Radius of the orbit, r₂ = 1.02R
- Time period, T₂ = 2π√((1.02R)³/GM) ------------ (2)
Percentage change in the time periods:
- Percentage change = (|T₁ - T₂|/T₁) * 100%
Substituting the values from equations (1) and (2):
- Percentage change = (|2π√(R³/GM) - 2π√((1.02R)³/GM)| / (2π√(R³/GM))) * 100%
- Simplifying the expression:
= (|√(R³) - √((1.02R)³)| / √(R³)) * 100%
= (|R - 1.02R| / R) * 100%
= (|-0.02R| / R) * 100%
= (0.02R / R) * 100%
= 0.02 * 100%
= 2%
Therefore, the percentage change in the time periods of the two satellites is 2%, which is closest to option D (3%).
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Community Answer
A satellite is launched into a circular orbit of radius R while a seco...
The time periods (T) of a satellite revolving around the earth of radius R and at height h is:
T = 2 x pi x [(R + h)3 / (G X M)]1/2
Hence, the time period is directly proportional to (R + h)3/2
From this, we have the ratio of the time periods as 1.03.
1.03 x 100 = 103. Hence, there is a 3% change in the time period.
T = 2 x pi x [(R + h)3 / (G X M)]1/2
Hence, the time period is directly proportional to (R + h)3/2
From this, we have the ratio of the time periods as 1.03.
1.03 x 100 = 103. Hence, there is a 3% change in the time period.
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A satellite is launched into a circular orbit of radius R while a second satellite is launched into an orbit of radius 1.02R. What is the percentage change in the time periods of the two satellites?a)0.7b)1c)1.5d)3Correct answer is option 'D'. Can you explain this answer?
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