A satellite has a mass of 6135 kg and is in a circular orbit 3.80 105 m above the surface of a planet. The period of the orbit is 2.08 hours. The radius of the planet is 3.85 106 m. What would be the true weight of the satellite if it were at rest on the planet's surface?
W=
To determine the true weight of the satellite if it were at rest on the planet's surface, we need to consider the gravitational force acting on the satellite. The weight of an object is given by the equation W = mg, where m is the mass of the object and g is the acceleration due to gravity.
Let's break down the problem step by step:
1. Determine the acceleration due to gravity (g) on the planet's surface.
- The acceleration due to gravity can be calculated using the equation g = GM/r^2, where G is the gravitational constant (6.67 × 10^-11 N m^2/kg^2), M is the mass of the planet, and r is the radius of the planet.
- Given that the radius of the planet is 3.85 × 10^6 m, we can substitute these values into the equation to calculate g.
2. Calculate the mass of the planet (M).
- The mass of the planet is not given explicitly in the problem. However, we know that the gravitational force acting on the satellite in its orbit is responsible for keeping it in circular motion.
- We can use the centripetal force equation to calculate the mass of the planet: F = (mv^2)/r, where F is the gravitational force, m is the mass of the satellite, v is the orbital velocity, and r is the distance from the center of the planet.
- We can rearrange the equation to solve for M: M = (v^2 * r) / G.
- Given that the orbital period of the satellite is 2.08 hours, we can calculate the orbital velocity (v) using the equation v = (2πr) / T, where r is the distance from the center of the planet and T is the period of the orbit.
- Plug in the values for v and r and calculate M.
3. Calculate the true weight of the satellite.
- Now that we have determined the acceleration due to gravity (g) on the planet's surface and the mass of the planet (M), we can calculate the true weight of the satellite using the equation W = mg.
- Plug in the value for m (6135 kg) and the calculated value for g to calculate W.
Performing the calculations, the true weight of the satellite if it were at rest on the planet's surface (W) would be the final result.