Saying that a centripetal force do no work over a mass moving on a circle is an error. However is an error easy to understand, once you see what is involved.

If you like physics then you know that the centripetal force do no work on a circle because it is always perpendicular to its trajectory. Search on google and you’ll note that everybody think the same. After all, if you check the work formula, it sounds right. Sorry, I am here to demonstrate that we were plain wrong.

Prior to any demonstration I should say that I know something about physics, I am aware of the $latex w=Fd \cos(\Theta)$ formula and I am (supossedly) in full possession of my mental faculties.

First let’s consider an “everyday” situation: It’s wendsday, you are on a spaceship traveling through the deep space when you find a 1000 Ton asteroid with a 2000km/h speed, and you feel creative today, so you decide to deflect that asteroid in a perfect circular 90° arc. From your physics knowledge you know that it will take energy to deflect it from its trajectory, but you know that if you deflect it always perpendicular to its trajectory then you will make no work over the asteroid and therefore deflecting it will cost you no energy. So, by using a strong cable, you start deflecting the asteroid in a big 90° arc. However, when you fynally stops, you find that the energy used is considerable. How can it be? You check the available data and double check the trajectory of the asteroid, just to find the trajectory was a perfect circle! Something somewhere must be wrong .

You call to your physics teacher, explain him the extrange results and he says everything is OK. You spent energy creating a force, and was that force what push the asteroid on a circle. Everybody knows that a force is required to make a circle, so was the centripetal force what created the circle, not the energy applied. That explains both where the energy was used, and the fact that a perpendicular force can’t do any work at all when the circle is created. Everybody is happy now! Everybody but you.

So you decide to check this “fact” by checking again the main computer data. You find that the force applied follows the equations

$latex F_{cx} = -F_c Sin \Theta$
$latex F_{cy} = -F_c Cos \Theta$

You also get the equations of displacement

$latex \Delta x = R\Delta\Theta Cos \Theta$
$latex \Delta y = -R\Delta\Theta Sin \Theta$

By the work formula, you find the work done is

$latex \Delta E_x =-V^2 m Cos\Theta Sin\Theta \Delta\Theta$
$latex \Delta E_y =V^2 m Sin\Theta Cos\Theta \Delta\Theta$

Wait a minute! That is clearly non zero work. There is positive work done in the y direction, and negative work done  in the x direction. So energy must be applied to make such work! Even negative work requires energy! This is nuts! We have been wrong about the work done over a circle for hundred of years!

Now, this clearly explains why took energy to deflect that asteroid. Even if the spaceship has low efficiency, is clear that some of the energy applied was used directly to do work over the asteroid. It is also clear why we don’t see any change in the asteroid energy: some additional energy was applied by the horizontal force to reduce the energy at the same time we were increasing it by applying the vertical force. Mistery solved!

Hey wait! If it takes energy to make a circle, why we see so many examples of circles that happend spontaneously? In fact, you can in theory attach the same asteroid to a big planet and no energy will be spent. How do you explain that?