Why Is Work Done by Gravity Zero for a Satellite in Orbit?

[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?

 

[Doc Al]

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

 

[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.

 

[Doc Al]

The force of gravity is pointing towards the center of the earth, at any moment; and the satellite is moving a circular path.

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

 

[Bashyboy]

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

 

[Doc Al]

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

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

 

[Bashyboy]

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

 

[Doc Al]

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

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.

 

[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?

 

[Doc Al]

If work had been done, the satellite would be speed up, right?

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

 

[Bashyboy]

Yes, you are right. Thank you very much!

 

[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?

 

[Doc Al]

What would happen if it were the case that the orbit was more elliptical. Would there be work done in that case?

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.)

 

[Bashyboy]

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

 

[Doc Al]

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

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.

 

[Bashyboy]

Excellent! Thank you!