Why do centrifugal force and centripetal acceleration have opposite signs?

Question

I know that rotating body has centripetal acceleration that is directed to the center of rotation.

$$a_{cp}=-\frac{v^2}{R}$$ (Lets imagine that body is located at 12 o'clock and that Ox axis is pointed from center to 12 o'clock, $a_{cp}$ is directed from body to center of the circle).

I know that according to Newton's second law there should be force that equals $$F=ma_{cp},$$ so there should be Force $$F_{cp} =-m\frac{v^2}{R}.$$ But from daily experience we know that centrifugal force that there is centrifugal force that acts on point that is located at 12 o'clock and it is parallel to line connecting center and point and is directed outward.

$$F_{cf}=m \frac{v^2}{R}$$

Why force that acts on body is directed to direction that is opposite to centripetal acceleration? What is solution of seeming contradiction with Newton's second law?

P.S. I know calculus and I still can't understand this? I spent a lot of time trying to understand this. UPDATE:

1)Let me sum up most essential of the answer. There is a way to determine acceleration by mental experiment. Imagine that that there is hollow box inside the body. In the box there is other smaller body that is attache to the springs. By measuring displacement of smaller body we can measure acceleration. If body is accelerating to the right smaller body will go to the left of the center of hollow box. If body is accelerating to the left smaller body will go to the right of the center of hollow box. If body is attached to the rope and is rotating than smaller body will move to direction opposite of center of rotation, showing that force is directed to the center of rotation. Direction of acceleration is the same as direction of the force.

2)I got from the link, that centripetal force is the tension force of the rope. https://www.khanacademy.org/science/physics/centripetal-force-and-gravitation/centripetal-forces/v/centripetal-force-problem-solving

Answer 1

Let's not think about circular motion for a minute. Let's think about driving a car in a straight line.

If you are in your car, and you slam on the accelerator, you feel like you're being pushed backwards into your seat. But of course, the actual force on you is pushing you forward! Similarly, if you slam on the brakes, you feel like you're being pushed forwards, even though the actual force on you is pushing you backwards.

Now lets turn the wheel. If you veer to the left, you feel like you're being thrown to the right, even though the net force on you is to the left (you end up turning left). Similarly, if you veer to the right, you feel like you're being thrown to the left, even though the net force on you is to the right.

This suggests a general rule. If you are in a closed box, and someone accelerates the box, you will feel some force on you. But you cannot tell the difference between the following two explanations for the force on you:

1. Someone has accelerated the box. Maybe the box is accelerating left, causing a leftward force on your feet, which makes you feel like you're falling to the right because there is no corresponding leftward force on your head.

2. The box is not accelerating at all. Someone has turned on some "force field" which is accelerating you to the right. You feel like you're falling right because something is pushing you to the right.

If we view the situation from the perspective of someone outside the box, they know we're in situation (1). The box is accelerating to the left, you are accelerating to the left with it, and you only feel like you're falling to the right because the floor is accelerating you leftward. We call the real force you're experiencing centripetal force.

If we are inside the box, though, we could equally well assume we're in situation (2). There is some mysterious force that is pushing us to the right. This force isn't "real". It's a made-up force that happens to explain everything we observe happening in the box if we assume the box itself isn't accelerating. We call this fictitious force the centrifugal force.

Answer 2

Imagine you're in a car, and someone is viewing the situation from a bird's eye view. You take a sharp left. Approximate the turn as perfectly circular. Even though you end up turning left, your body feels as though it "wants" to go right, outwards. After all, if you're leaned against the door, you will feel the door exert a force on you. The force would be the normal force $N$, providing the necessary centripetal force pushing you INWARDS to keep going in that circle (also friction, but we can ignore that for this purpose).

Now, let's pick a non inertial frame of reference where you're not accelerating. That would be right inside your car. Since you're not accelerating, the net force must be zero. But the door is pushing on you with some force $N$ to the left. But you're not accelerating in that frame, therefore there must be some other force to balance it out, pushing you to the right. That "other force" is the fictitious centrifugal force.

So, in circular motion, in an inertial frame there will be a force pushing you towards the inside of the circle. The vector sum of the forces directed into (or out of) the center is the centripetal force. Now, in our chosen non-inertial reference frame such that the net force is zero, something must counteract that centripetal force that pushes you inside. That something is the fictitious centrifugal force.