# Do centripetal and reactive centrifugal forces cancel each other out?

In order for a body to move with uniform velocity in a circular path, there must exist some force towards the centre of curvature of the circular path. This is centripetal force. By Newton's Third Law, there must exist a reactive force that is equal in magnitude and opposite in direction. This is the reactive centrifugal force.

My question is simple, and it is probably the result of lack of common sense but here it goes: In uniform circular motion, why don't these forces simply cancel each other out? If they did, how would we know they exist in that situation?

When I swing a rock tied to a rope, I feel the centrifugal force, but not the centripetal force. In this situation how can the reactive force be greater than the force itself?

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–  Ben Crowell Oct 26 '14 at 20:08

Some of the confusion about centripital, centrifugal, and reactive forces is just vocabulary. It can be easier to understand if we consider a similar example without rotation.

Suppose you are floating in space near a rocket. A rock is tied to the rocket with a thread. When the engine starts, the rocket pulls on the thread and exerts a force on the rock. The rock accelerates.
$T = m_{rock} * a_{rock}$

The reaction force is the equal and opposite force the rock exerts on the rocket. The rock pulls on the thread and reduces the acceleration of the rocket.
$F = F_{rocket} - T$
The reaction force doesn't cancel anything. It is just a force that added to all the other forces on the rocket.

Since you are floating near the rocket, you see the rocket move. The pilot seated in the rocket finds it more convenient to adopt point of view where the rocket stays at rest. At time $t_0$, the seat is right under him. At time $t_1$, the seat is still right under him.

For the pilot to use laws like $F = ma$, he must redefine acceleration so that $a = 0$. This means he must redefine force so that $F = 0$.

The changes in definition are not big. Everything is consistent if he adds a fictitious acceleration $a_{fict}$ to all accelerations, where $a_{fict} = -a_{rock}$. He adds a fictitious force $f_{fict} = ma_{fict}$ to all forces.

He is saying "$f_{fict}$ acts on everything in the universe, causing everything to accelerate backword with acceleration $a_{fict}$. The total acceleration of rocket+rock is $0$ because of the additional force $F$ from the engine.

Fictitious forces do not cancel anything. They just change your point of view, or frame of reference.

Returning to circular motion, suppose you are floating near a rocket that is stationary but spinning. A rock is tied to the rocket, and rotates around the rocket.

The rocket engine provides the centripetal force that keeps the rock moving in a circle.

The reaction force holds the rocket stationary.

Centrifugal force is useful to an ant on the rock. The ant finds it useful to adopt a frame of reference where the rock is stationary. This is a more complex case, because $a_{centrifugal}$ depends on position.

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NO, They do not cancel out each other, while centripetal (center seeking force) is generally provided by some other agency/force, like for revolution of planets it is provided by gravitational force, centrifugal force(outward force) is a pseudo force which is felt in the reference frame of the revolving/rotating body. Clearly since the two forces belong in different frames, they do not cancel out each other in your frame i.e. from the viewers frame they cancel out only in the frame of reference of body as the body does not move in that frame.

When you are rotating a stone/ball tied to a thread you seem to think that you are feeling an outward/centrifugal forcre, but it is actually the tension of the thread, see at the end of the ball tension is directed towards the centre of rotation and is hence centripetal force, but the same tension at the point/centre of rotation is directed towards the ball, therefore you feel an outward force but it is not centrifugal force.

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In order for a body to move with uniform velocity in a circular path, there must exist some force towards the centre of curvature of the circular path. This is centripetal force. By Newton's Third Law, there must exist a reactive force that is equal in magnitude and opposite in direction.

True, although the adjective "reactive" is meaningless. There is no distinction between forces that are "reactive" and forces that aren't.

This is the reactive centrifugal force.

No. Centrifugal force refers to one of the fictitious forces that acts on objects when you describe everything in a non-inertial, rotating frame. It's not a real force. "Fictitious" means fake or fictional. One way to tell that fictitious forces are not real is that unlike real forces, they are not exerted by an object on another object. This means that Newton's third law doesn't apply to them. In general Newton's laws cannot be made to hold in a noninertial frame.

In uniform circular motion, why don't these forces simply cancel each other out?

In a description based on an inertial frame of reference, the real centripetal force could be the only force acting on the object. There is no fictitious centrifugal force, and nothing cancels. For example, if a whirl a rock on the end of a string, then the string's normal force on the rock is the only force on the rock (neglecting gravity). There is a third-law partner to this force, which is the rock's normal force on the string. As always with third-law partners, these two forces act on different objects, and therefore it doesn't make sense to talk about their cancellation -- cancellation implies addition, but it doesn't make sense to talk about adding forces that act on different objects.

In a description using a frame of reference that rotates along with the rock, there is a fictitious centrifugal force and this force does cancel the string's force on the rock. These two forces are not third-law partners.

When I swing a rock tied to a rope, I feel the centrifugal force, but not the centripetal force.

No, the centrifugal force would be a fictitious force acting on the rock if you chose to adopt a rotating frame. (In that frame, the centrifugal force acting on you would be zero, since you're at $r=0$.) The force you feel is not a fictitious centrifugal force. It's a real normal force of the rope on your hand.

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The term centripetal force is only relevant in an inertial frame. However it is always created by some physical force, gravitation or the friction from a seat or something. In the rotating system, the fictional centrifugal force has to be invoked. The physical force that makes up the centripetal force also exist in the rotating frame and here it is exactly countered by the centrifugal force. So my answer is YES.

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Action and reaction pairs act on two different objects, so it's not clear how they will cancel each other out.

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This looks like you're asking a new question. –  Kyle Kanos Jan 5 at 12:52
@KyleKanos: I think in a way he was being sarcastic but I do think he's stumbled onto a gem of a question there. I have no clue on how to phrase it properly but I think some good curiosity can be salvaged from this. Nevertheless, this answer is not at par with a decent answer. Hence, I've flagged it. –  Nick Jan 5 at 14:01