# Homework Help: Finding Work Done By Gravity

1. Dec 18, 2012

### Bashyboy

Okay, the problem is:

A $1200~kg$ satellite is in a circular orbit around the Earth at a radius of $8.77\cdot 10^6~m$. What is the work done by gravity when the satellite has gone a quarter of the war around its orbit? The acceleration due to gravity at the height of the satellite is $5.19\cdot ~m/s^2$

Well, I figured, that since calculation involves the satellite not making a full rotation (a zero displacement, the work done done would be a nonzero value. Apparently, though, this isn't true, and it is indeed zero. Why is this true?

2. Dec 18, 2012

### Staff: Mentor

What direction is the force of gravity at any moment? In what direction is the satellite moving at any moment? Is work being done?

3. Dec 18, 2012

### Bashyboy

The force of gravity is pointing towards the center of the earth, at any moment; and the satellite is moving a circular path. I figured that the force of gravity was being applied over that distance, thus doing work.

4. Dec 18, 2012

### Staff: Mentor

Good. What is the angle between the force of gravity and the satellite's velocity?

5. Dec 18, 2012

### Bashyboy

The angle between the instantaneous velocity and force of gravity is 90 deg.

6. Dec 18, 2012

### Staff: Mentor

Right. So is any work done if the velocity is always perpendicular to the force?

7. Dec 18, 2012

### Bashyboy

Well, apparently by the mathematical definition of work there is no work being done. Is there any physical reasoning to this, or no?

8. Dec 18, 2012

### Staff: Mentor

Not sure what you mean by 'physical reasoning'.

If you keep pulling something that continually moves sideways, no work is done and the speed of the object remains unchanged. Another example is twirling a ball on the end of a string in a horizontal circle. The tension in the string always acts sideways to the ball's motion, so no work is done.

9. Dec 18, 2012

### Bashyboy

Ah, yes. I forgot that tiny detail: the orbit of the satellite is at a constant speed, provided that it has no way of propelling itself. If work had been done, the satellite would be speed up, right?

10. Dec 18, 2012

### Staff: Mentor

Right. (Or slow down, depending on the sign of that work.)

11. Dec 18, 2012

### Bashyboy

Yes, you are right. Thank you very much!

12. Dec 19, 2012

### Bashyboy

I have another question. In the problem, we assume that the orbit is perfectly spherical. What would happen if it were the case that the orbit was more elliptical. Would there be work done in that case?

13. Dec 19, 2012

### Staff: Mentor

Sure. Gravity does positive work as the satellite gets closer to earth and negative work as it gets farther away. (The net work will be zero in an elliptical orbit.)

14. Dec 19, 2012

### Bashyboy

And is that why the satellite speeds up as it follows the part of the elliptical orbit that is near earth?

15. Dec 19, 2012

### Staff: Mentor

Exactly.

You can also think of it in terms of mechanical energy being conserved. As the satellite gets nearer the earth, the gravitational PE decreases and the kinetic energy increases.

16. Dec 19, 2012

### Bashyboy

Excellent! Thank you!