A satellite has a mass of 6090 kg and is in a circular orbit 4.63 × 105 m above the surface of a planet. The period of the orbit is 2.4 hours. The radius of the planet is 4.86 × 106 m. What would be the true weight of the satellite if it were at rest on the planet’s surface?
Thanks
To find the true weight of the satellite if it were at rest on the planet's surface, we need to calculate the gravitational force acting on it. The formula for the gravitational force is:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two interacting objects, and r is the distance between their centers of mass.
In this case, we want to find the gravitational force between the satellite and the planet when the satellite is at rest on the planet's surface. Assuming the satellite's mass remains the same, we need to find the mass of the planet. We can use the given information to calculate it.
Given:
Satellite mass, m1 = 6090 kg
Orbit radius, r = 4.63 × 10^5 m
Planet radius, R = 4.86 × 10^6 m
Period of orbit, T = 2.4 hours = 2.4 * 60 * 60 seconds
First, let's calculate the mass of the planet using the period of orbit:
Velocity of the satellite in its orbit can be calculated using the formula:
v = (2 * pi * r) / T
where v is the velocity of the satellite.
Substituting the given values:
v = (2 * pi * (4.6 * 10^5 m)) / (2.4 * 60 * 60 s)
v ≈ 6410 m/s
Now, we can use the formula for centripetal force to calculate the mass of the planet:
F_c = (m1 * v^2) / r
Since the gravitational force and the centripetal force are equal in magnitude:
G * (m1 * m2) / r^2 = (m1 * v^2) / r
Simplifying:
m2 = (v^2 * r) / (G * r)
Substituting the given values:
m2 = (6410^2 m^2/s^2 * (4.6 * 10^5 m)) / (6.67430 × 10^-11 N m^2/kg^2 * (4.63 * 10^5 m))
m2 ≈ 1.716 * 10^24 kg
Now that we have the mass of the planet, we can calculate the gravitational force on the satellite when it is at rest on the planet's surface:
F = G * (m1 * m2) / R^2
Substituting the given values:
F = (6.67430 × 10^-11 N m^2/kg^2 * (6090 kg) * (1.716 * 10^24 kg)) / (4.86 * 10^6 m)^2
F ≈ 2.07 * 10^6 N
Therefore, the true weight of the satellite if it were at rest on the planet's surface would be approximately 2.07 * 10^6 N.