A satellite has a mass of 6179 kg and is in a circular orbit 4.83 × 105 m above the surface of a planet. The period of the orbit is 2.4 hours. The radius of the planet is 4.03 × 106 m. What would be the true weight of the satellite if it were at rest on the planet’s surface?
To find the true weight of the satellite on the planet's surface, we need to calculate the gravitational force acting on it.
The formula for 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 m^3 kg^(-1) s^(-2)
m1 and m2 are the masses of the two objects (in this case, the satellite mass and the planet mass)
r is the distance between the centers of the two objects (in this case, the radius of the planet)
First, we need to calculate the mass of the planet using the radius and the formula for the volume of a sphere:
V = (4/3) * π * r^3
Since density is mass divided by volume, we can rearrange the formula to solve for mass:
M = (4/3) * π * r^3 * D
Where:
M is the mass of the planet
D is the density of the planet
Now that we have the mass of the planet, we can proceed to calculate the gravitational force between the satellite and the planet.
F = (G * m1 * m2) / r^2
Where:
F is the gravitational force
m1 is the mass of the satellite
m2 is the mass of the planet
r is the distance between the center of the planet and the satellite (radius of the planet + height of the satellite above the surface)
Finally, the weight of an object is the force acting on it due to gravity, so the true weight of the satellite on the planet's surface would be the gravitational force acting on the satellite.
I will now calculate the true weight of the satellite using the given information:
m1 = 6179 kg (mass of the satellite)
r = 4.03 × 10^6 m (radius of the planet)
h = 4.83 × 10^5 m (height of the satellite above the surface)
T = 2.4 hours (period of orbit)
1. First, we need to convert the period of orbit from hours to seconds:
T = 2.4 hours * 3600 seconds/hour
2. Next, we need to calculate the velocity of the satellite in its circular orbit using the formula:
v = 2πr / T
3. Then, we can calculate the distance from the center of the planet to the satellite:
d = r + h
4. Now, we can calculate the mass of the planet using the formula for the volume of a sphere and the known density of the planet.
5. Finally, we can calculate the true weight of the satellite using the formula for gravitational force.
Let me do the calculations.