# The centripetal force work problem – solved!

A few weeks ago, I wrote the article “Does the centripetal force do some work?“. Since then, I keep myself searching for an answer. Last week, the right answer finally arrived to my inbox. And the answer came from Henry Reich, the creator of MinutePhysics. To be honest, when I wrote him the email my chances of get an answer were really low. However, to my surprise, Henry was patient enough to show me where my error was. Here is the transcription (simplified) of the problem I exposed to Henry, and his very clear response.

First, I explain him why I thought the centripetal force does work:

Henry,
Let me start saying I have some formal physics instruction so I am aware of the energy conservation laws. Actually I don’t think this law can be easily broken. I also know that there is no work done when making a circular path, because of the Work=F*d*Cos(angle) formula.

Now, what takes a man aware of the physics laws to think that it requires work to make a circle? The problem of deflecting an asteroid.

The “deflect an asteroid” problem:
Imagine an asteroid, 1 million tons weight, traveling the flat spacetime with velocity Vx (lets think of it as traveling on a x-y plane). Please note there is no string attached to the asteroid. Just an asteroid happily traveling in a straight line.

You can exert a force Fy over the asteroid, and that force will deflect its trajectory, so Fy will do a work Wy=Fy*dy .
or
You can exert a Fx over the asteroid, opposed to its velocity, so you can slow down the asteroid, and the work done will be Wx=-Fy*dx (negative work, by convention).
To this point every physicist agree there is work done in each of these cases. Perfect!

Now, what if we apply both forces simultaneously? For most physics, work done disappears. Why? Because a circle does not require energy to be created. I know that. The force is always perpendicular to the trajectory, so no work is done.

There is a force Fy applied simultaneously with a Fx force, the asteroid deflects (more than when the Fy force was applied alone), and still there is no work done. Don’t you think this sounds odd, just a little?

If this is true, we can actually deflect such a big asteroid in a circular path, and it will use no energy to do that, because no work has been done!
We can even literally stop the asteroid on a 1 meter circle without any energy expended! I don´t think that could happen in a real world.

Of course, many argue that in fact we will require lots of energy to generate the forces, and that can be true. However, the energy required to generate the force is not considered in my calculation, just the net work done over the asteroid by the force applied. If it takes energy to generate the force, that will add to the net energy required to deflect the asteroid. And that means even more energy would be required.

Does this makes sense?

Please note this goes against what we have learned (It took me months to start to believe this could be true). Also note I am aware that there are lots of things rotating forever with zero energy required (I will explain this later, if we find the asteroid case makes sense).
All we have learned to this day say a circle requires no energy, but please, don’t discard this possibility too fast. Let’s imagine the asteroid and ask your self -no equations involved- Can we really deflect an asteroid without work?

Thanks again for your time reading this.

Henry’s answer to this problem was, of course, brilliant. This is the only response that has taken two apparently different realities and generated a coherent response:

Stalin,

I think you are putting too much faith in your intuition and not enough in your logical reasoning.

Also, I think you are confusing what is meant by “work”: you’re totally correct that in certain cases it can take the expenditure of a LOT of energy to keep an object rotating in a circle – moving an asteroid with rockets is one example – but that is not what work is. “Work” in physics parlance, means specifically energy ADDED to the system in question, and while the rockets may expend a lot of energy changing the asteroid’s direction of motion so that it travels in a circle, if they do so entirely by firing perpendicularly then they do not change the asteroid’s speed or kinetic energy – just the direction of the velocity. The point is, if they stopped firing and let the asteroid continue on its path, it would have the exact same energy it started with (though be traveling in a new direction). And in physics parlance, this means that zero work has been done on the asteroid. This doesn’t mean that nothing happened, or that no one expended any energy! It just means that the energy OF THE ASTEROID has not changed.

I think that you are kind of intuitively using a different definition for “work” than what is meant by physicists. All physicists mean by “work” is the change of total energy of an object over time. If an asteroid in free space starts with velocity V and that velocity changes direction but not magnitude (and nothing else changes), then the total energy of the asteroid is still 1/2*m*V^2 and so the change of energy is zero. If you want “work” to mean something different, that’s fine, but you can’t then go argue with physicists that they’re wrong saying “no work happens”, because by their definition, they are right.

This is why I hate the use of the word “work” by physicists. If I could abolish that word forever and just have everyone say “change of energy” then I think all sorts of confusion would be avoided, because the word “work” has many colloquial meanings, many of which are closer to “power” or “force” or other things than “change of energy”.

Henry