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Explanation of the centripetal force and acceleration

I just need a detailed explanation of the centripetal force and acceleration. Please take it step by step so that I can understand
Centripetal force and centripetal acceleration are linked by Newton's 2nd law, F = ma. Mass is a scalar, and the a and F vectors point in the same direction. When an object moves in a circle with constant speed, the motion is called uniform circular motion. The speed, which is the magnitude of v, is constant, but the velocity vector v is not, since the object is always changing its direction.

Since velocity is changing, there must be an acceleration. This acceleration is special, because it preserves the speed (magnitude of v) and changes the object's motion just enough so that it moves in a circle.

It can be proven that this acceleration vector always points to the center of the circle, and that its magnitude is v2/r. The centripetal force, by Newton's 2nd law, has magnitude mv2/r.

That is a basic introduction that echos what is in a lot of introductory textbooks. You'll need to tell us exactly what you have trouble understanding for us to really be able to help you.
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What many students don't understand or have trouble with at first is when they try to imagine themselves being in a situation where a centripetal force is acting on them.

For example, in a theme park ride, imagine a circular track with a fast 'train' going in circles.
If you are in the train, the force you seem to be feeling is pulling you out of the circle, you are pushed against the outer side of the train.

Since you feel a force pushing you 'out of the circle', you may think that the centripetal force is also directed 'out of the circle'.
However, this is NOT the case!

The centripetal force is actually pointing directly to the center of the circle (so to the inside of the circle).

Because the theory of centripetal force and the practise of being pushed out of the circle seems to contradict eachother many students are puzzled here...

The explanation is as follows:

Whenever an object changes its speed or direction (it undergoes an acceleration), there MUST be a force acting on the object. This is Newton's first law: If no forces act on a body, it cannot undergo any acceleration and thus remains in rest or remains travelling in a constant speed and direction.

From this we can now see that there MUST be a force pulling the train into the circle. If this force was not present, according to newton's first law, the train can NOT have an acceleration (in this case, change of direction) and thus cannot go in a circle.

The direction of the force follows from Newton's second law as stated in the post above. The direction of the acceleration must be the same as the direction of the force that creates the acceleration.
In this case, the acceleration is towards the center of the circle (how else could the train make the bend to complete the circle?) so the force must also be directed towards the center.
This force is called the centripetal force.

The force you feel that seems pulls you out of the train is not actually a force. It is what we call a fictitious force. There are a few ways to explain the apparent 'fake' force and I think the most easiest is to look at Newton's first law again.

If a body is moving at some speed, it is 'hard' to make the body stop, go faster or go in a circle. Consider yourself in a car. What happens if you suddenly brake? You lunge forward.
You could see this as if any body with some speed 'does not like to' change its speed.
If you are in the train from above, you are travelling at quite a large speed and therefore 'do not like' to change direction. Instead, you 'try' to remain in your current path, which is to keep going forward.
Because the train moves into the circle, it kinda bends into you, which is why it seems you are being pulled out.
(This force is called the centrifugal force)

The physical explanation is a bit harder. It involves the fact that Newton's laws are only valid in an inertial frame of reference (basically, a frame of reference with no acceleration). Since the train is going in a circle, it has an acceleration and technically, Newton's laws are not valid.
To make them valid once again however, you can do a change of coordinates from an inertial coordinate system to a rotating coordinate system. Doing this alters the equations of Newton's laws and a few 'mass x acceleration' (force) terms 'appear'.
These apparent forces are NOT real and also do not apply to Newton's third law (action = - reaction), there is no reactional force to fictitious forces.

Last thing... The centripetal force is not directly acting on you. It is acting on the train. The train is connected to some rails which steer it into the circle.
When the train starts to bend into the circle however, the friction between the seat and you (or possibly the 'wall' of the train pushing against you) is what steers you in the circle aswell.

If the seat was completely frictionless and there was no wall (and nothing to hold on) you could not possibly stay in the train. You would slide as as soon as it starts to move; you would go in a straight line, because there is NO force acting on you!
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