A 7450 kg satellite has an elliptical orbit, as in Figure 6.9b. The point on the orbit that is farthest from the earth is called the apogee and is at the far right side of the drawing. The point on the orbit that is closest to the earth is called the perigee and is at the far left side of the drawing. Suppose that the speed of the satellite is 2730 m/s at the apogee and 8700 m/s at the perigee.
Find the work done by the gravitational force when the satellite moves from the apogee to the perigee.
Find the work done by the gravitational force when the satellite moves from the perigee to the apogee.
To find the work done by the gravitational force when the satellite moves from the apogee to the perigee, we can use the formula for work:
Work = Force * Distance * cos(theta)
In this case, the force is the gravitational force between the satellite and the Earth, the distance is the distance between the apogee and the perigee, and theta is the angle between the force and the displacement.
First, we need to find the distance between the apogee and the perigee. In an elliptical orbit, this can be found by subtracting the radius of the Earth from the sum of the apogee altitude and the perigee altitude.
Let's say the apogee altitude is h1, and the perigee altitude is h2. Then the distance between the apogee and the perigee is:
Distance = 2 * (h1 + h2)
Next, we need to calculate the gravitational force at the apogee and the perigee. The gravitational force can be calculated using Newton's law of universal gravitation:
Force = (G * m1 * m2) / r^2
Where G is the gravitational constant, m1 and m2 are the masses of the two objects (in this case, the satellite and the Earth), and r is the distance between them (in this case, the distance between the satellite and the center of the Earth).
Once we have the values for the force at the apogee and perigee, we can substitute them into the work formula along with the calculated distance between the apogee and perigee to find the work done by the gravitational force. Since the force and displacement are in the same direction (towards the center of the Earth), the angle theta is 0 degrees, so cos(theta) = 1.
Once we have the work done from the apogee to the perigee, we can repeat the steps above to find the work done from perigee to apogee. The only difference would be in the values of speed and altitude at the two points.
I hope this helps you understand how to calculate the work done by the gravitational force in a satellite's orbit.