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Imagine an object moving in a straight line in space, perhaps shaped like a yo-yo (to make rotational effects negligible). Suddenly a constant force is exerted at right angles to the direction of motion, such that the object is now deflected into a circular path.

What is the energy required (or work done) to maintain the constant deflection of this circular path?

Start over, and imagine a second identical object moving in the opposite direction, but offset so that when a similar force is applied to it, it travels in the same circular path, but offset by 180 degrees. Clearly, the energy required to keep both objects moving in the path is twice that for just one of them. Now, start again, but this time have the two objects instead link up with a cable and snap fastener, as they go by each other. The cable tension causes both objects to travel in the same circular path as before.

Does the cable do the same work that was done before to keep the two objects in the circular path? If not (and besides, how can a cable do work?), then was the energy wasted before, to do what a simple cable can do without energy?

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Zero since the force is perpendicular to the motion. There is not distance traveled in the direction of the force and thus not work done (or needed). – ja72 Mar 9 '13 at 6:34
mistyped comment removed – curiousguy Mar 9 '13 at 6:37
If there is no work done or needed, then we can happily be safe from incoming asteroids (no matter how massive) since no work is needed to deflect them. – curiousguy Mar 9 '13 at 6:38
Yes you are confused indeed. With statements like "clearly it takes twice the energy". Demonstrate it before you state it because it is wrong. – ja72 Mar 9 '13 at 6:51
Somewhat hard to demonstrate without pictures, perhaps you misunderstand. Imagine you are a little tugboat in space, and for some reason you want to tug an incoming asteroid into a circular path. Do you believe you can do with without constantly using energy? And whatever energy it takes, would be doubled if two tugboats deflected two asteroids. – curiousguy Mar 9 '13 at 6:58

I think what is confusing you is that you have the idea of, for example, a rocket motor blasting away to apply the force, and the rocket motor is clearly expending energy. You correctly say that if you could tie your asteroid to something heavy with a cable then it would move in a circle without any energy being expended. So how come the rocket requires energy but the cable doesn't?

The solution is that the rocket motor is indeed doing work, but it's not doing work on the asteroid, it's doing work on the fuel that it's blasting out i.e. the energy it expends is being used to accelerate its exhaust gases from zero to some high speed.

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