A 21.0-kg satellite has a circular orbit with a period of 2.37 h and a radius of 8.50×106 m around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is 7.10 m/s2, what is the radius of the planet?
I am not sure which formula to use to find the mass and then radius of the planet :/
oops, typo haha...nevermind, I found your answer below. thanks!!!
A 20 kg satellite has a circular orbit with a period of 2.9 h and a radius of 9.6 ✕ 106 m around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is 8.1 m/s2, what is the radius of the planet?
To find the radius of the planet, we can use the following steps:
1. Using the period of the satellite's orbit, we can calculate the gravitational constant of the planet.
The period of the satellite's orbit is given as 2.37 hours, which can be converted to seconds by multiplying by 3600 (60 seconds in a minute, 60 minutes in an hour). Therefore, the period of the satellite's orbit is 2.37 x 3600 = 8520 seconds.
2. Using the radius of the satellite's orbit and the gravitational constant, we can calculate the mass of the planet.
The gravitational constant can be calculated using the formula:
G = (4π²r³) / (T² * M)
Where G is the gravitational constant, π is pi (approximately 3.14), r is the radius of the satellite's orbit, T is the period of the satellite's orbit, and M is the mass of the planet.
We can rearrange the formula to solve for M:
M = (4π²r³) / (G * T²)
Substituting the given values: r = 8.50 x 10^6 m and T = 8520 s, and using the known value for the gravitational constant G = 6.67 x 10^-11 m³/(kg·s²), we can calculate the mass:
M = (4π²(8.50x10^6)³) / (6.67 x 10^-11 * (8520)²)
3. Finally, with the mass of the planet, we can calculate the radius of the planet using the gravitational acceleration on its surface.
The radius of the planet can be calculated using the formula:
g = (G * M) / R²
Where g is the gravitational acceleration on the planet's surface, G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.
We can rearrange the formula to solve for R:
R = √((G * M) / g)
Substituting the known values: G = 6.67 x 10^-11 m³/(kg·s²), M is the calculated mass of the planet, and g = 7.10 m/s² (given), we can calculate the radius of the planet:
R = √((6.67 x 10^-11 * M) / 7.10)
By following these steps and using the appropriate formulas, you will be able to find the radius of the planet.