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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. 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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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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