Finding radius of satellite in orbit given mass and rotational period

In summary, to place a spy satellite in geosynchronous orbit above Planet X, with a mass of 5.90 x 10^24 kg and a rotational period of 26.4 hours, the radius of the satellite's orbit should be 2.06 x 10^8 meters or 2.06 x 10^5 kilometers, and the orbital speed would be 2.88 km/s. This was calculated using the law of periods, rearranged to solve for radius, and plugging in the given values for G, M, and T. The discrepancy in the previous attempt was due to a mistake in entering the numbers into the calculator.
  • #1
csgirl504
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Homework Statement



Suppose that Captain Omega of the Imperial Space Patrol wishes to place a spy satellite in geosynchronous orbit above the mysterious Planet X, which has a mass of 5.90 x1024 kg, and a rotational period of 26.4 hours.

(a) What should be the radius of the satellite's orbit?
(b) What would be the orbital speed of the satellite?

Homework Equations


The Attempt at a Solution



I used the law of periods : T2/r3 = 4∏2/GM

So I rearranged that to get:

r = (GMT2 / 4∏2) 1/3

I plugged in:
G = 6.67 e -11
M = 5.9 e 24 kg
T = 95040s

I ended up with 2.06e8 m = 2.06e5 km

But that answer isn't right and I don't understand what I'm doing wrong.
 
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  • #2
I plug in your numbers to your formula and I get a different answer. You must have forgot a step in your arithmetic.
 
  • #3
Thank you! I'm not sure what I was doing wrong..mistake while entering in the calculator I guess!
 

1. What is the formula for finding the radius of a satellite in orbit given its mass and rotational period?

The formula for finding the radius of a satellite in orbit is r = (G * M * T^2) / (4 * π^2), where r is the radius in meters, G is the gravitational constant (6.67 x 10^-11 N*m^2/kg^2), M is the mass of the planet in kilograms, and T is the rotational period of the satellite in seconds.

2. Can the radius of a satellite in orbit be calculated if the mass and rotational period are not provided?

No, the radius of a satellite in orbit cannot be calculated without knowing its mass and rotational period. These two factors are essential in determining the gravitational force and centripetal force acting on the satellite, which ultimately determines its orbital radius.

3. How does the mass of the planet affect the radius of a satellite in orbit?

The mass of the planet directly affects the radius of a satellite in orbit. The greater the mass of the planet, the stronger the gravitational force and the larger the orbital radius of the satellite will be. This can be seen in the formula, where the radius is directly proportional to the mass of the planet.

4. Why is the rotational period of the satellite important in calculating its orbital radius?

The rotational period of the satellite is important because it determines the speed at which the satellite is orbiting the planet. This speed, along with the gravitational force, is used to calculate the centripetal force acting on the satellite and ultimately determine its orbital radius.

5. How does the orbital radius of a satellite affect its stability in orbit?

The orbital radius of a satellite directly affects its stability in orbit. If the satellite is too close to the planet, it will experience a stronger gravitational force and may crash into the planet. On the other hand, if the satellite is too far away, the gravitational force may not be strong enough to keep it in orbit and it may drift away. Finding the correct orbital radius is crucial for a satellite to maintain a stable orbit around a planet.

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