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A satellite is launched into a circular orbit of radius R around earth, while a second satellite is launched into a circular orbit of radius 1.02R. The percentage difference in the time periods of the two satellites is:
- a)1.5
- b)2.0
- c)3.0
- d)0.7
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
A satellite is launched into a circular orbit of radius R around earth...
T2 ∝ R3
T = kR3/2
Percentage change = 3%
T = kR3/2
Percentage change = 3%
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Community Answer
A satellite is launched into a circular orbit of radius R around earth...
Explanation:
Given:
Radius of first satellite's orbit = R
Radius of second satellite's orbit = 1.02R
Formula for the time period of a satellite:
The time period of a satellite in a circular orbit is given by the formula:
T = 2π√(r³/GM)
where T is the time period, r is the radius of the orbit, G is the gravitational constant, and M is the mass of the Earth.
Finding the time period of the first satellite:
Using the above formula, the time period of the first satellite with radius R is:
T₁ = 2π√(R³/GM)
Finding the time period of the second satellite:
Using the same formula, the time period of the second satellite with radius 1.02R is:
T₂ = 2π√((1.02R)³/GM)
Calculating the percentage difference:
The percentage difference between the two time periods can be calculated using the formula:
Percentage Difference = |(T₂ - T₁)/T₁| * 100
Substituting the values of T₁ and T₂ in the above formula:
Percentage Difference = |(2π√((1.02R)³/GM) - 2π√(R³/GM))/ (2π√(R³/GM))| * 100
Percentage Difference = |(√((1.02R)³/GM) - √(R³/GM))/ √(R³/GM)| * 100
Simplifying the above expression:
Percentage Difference = |(√(1.02³R³) - √(R³))/ √(R³)| * 100
Percentage Difference = |(1.02 - 1)/1| * 100
Percentage Difference = |0.02/1| * 100
Percentage Difference = 2%
Therefore, the percentage difference in the time periods of the two satellites is 2%. Hence, the correct answer is option C.
Given:
Radius of first satellite's orbit = R
Radius of second satellite's orbit = 1.02R
Formula for the time period of a satellite:
The time period of a satellite in a circular orbit is given by the formula:
T = 2π√(r³/GM)
where T is the time period, r is the radius of the orbit, G is the gravitational constant, and M is the mass of the Earth.
Finding the time period of the first satellite:
Using the above formula, the time period of the first satellite with radius R is:
T₁ = 2π√(R³/GM)
Finding the time period of the second satellite:
Using the same formula, the time period of the second satellite with radius 1.02R is:
T₂ = 2π√((1.02R)³/GM)
Calculating the percentage difference:
The percentage difference between the two time periods can be calculated using the formula:
Percentage Difference = |(T₂ - T₁)/T₁| * 100
Substituting the values of T₁ and T₂ in the above formula:
Percentage Difference = |(2π√((1.02R)³/GM) - 2π√(R³/GM))/ (2π√(R³/GM))| * 100
Percentage Difference = |(√((1.02R)³/GM) - √(R³/GM))/ √(R³/GM)| * 100
Simplifying the above expression:
Percentage Difference = |(√(1.02³R³) - √(R³))/ √(R³)| * 100
Percentage Difference = |(1.02 - 1)/1| * 100
Percentage Difference = |0.02/1| * 100
Percentage Difference = 2%
Therefore, the percentage difference in the time periods of the two satellites is 2%. Hence, the correct answer is option C.
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A satellite is launched into a circular orbit of radius R around earth, while a second satellite is launched into a circular orbit of radius 1.02R. The percentage difference in the time periods of the two satellites is:a)1.5b)2.0c)3.0d)0.7Correct answer is option 'C'. Can you explain this answer?
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