What happens to net acceleration in a non-inertial reference frame with cenripetal force?

In a non-inertial reference frame, the centripetal force is balanced by a centrifugal force. Therefore, shouldn't the net acceleration be zero? Yet, you still feel an acceleration outwards? Why is that?

Recently you have posted three questions, (Why do we feel a force in circular motion? and If an object moving in a circle experiences centripetal force, then doesn't it also experience centrifugal force, because of Newton's third law? and this question), relating to non-inertial frames of reference.

The first thing to note is that any centrifugal/pseudo/fictitious/inertial/d'Alembert force is not a real force, rather it is introduced for the convenience of being able to use Newton’s second law in a non-inertial frame of reference. There is no Newton third law pair to this introduced force.

In a non-inertial reference frame, the centripetal force is balanced by a centrifugal force. Therefore, shouldn't the net acceleration be zero?

If those two "forces" are equal in magnitude and opposite in direction then in that particular non-inertial frame the acceleration is zero.

Yet, you still feel an acceleration outwards?

The operative word here is "feel." What you don't feel is an outward acceleration. What you do feel is being stretched by two "forces".

Imagine that you have an object moving at constant speed in a circle because a string is pulling on it.

Standing on the ground (inertial frame of reference) you reason using Newton's second law, that there is a force on the object due to the string which is causing a centripetal acceleration of the object.

Now you sit on the object and you observe that the object is not moving relative to you ie its acceleration in this frame of reference is zero, and yet the string is still exerting a force on the object.

So obviously Newton's second law is not true.

In this example to be able to use Newton's second law in this situation a force, equal in magnitude and opposite in direction to the force exerted on the body by the string, is introduced.

This is sometimes called the centrifugal force and resolves the problem of Newton's second law not working in that now the net force on the object is zero which means that it has zero acceleration.

This introduced force is not "due to" anything and so does not have a Newton third law pair.

As soon a you start mentioning "what you feel" you have to be careful.

When you are being stretched you associate the stretching of your body with two external forces at opposite ends of your body trying to pull you apart and the opposite is true when you are being compressed.

You are the system under consideration.

Suppose that you are sitting in a seat and being accelerating relative to the ground in a forward direction by an external force, also in the forward directed, exerted by the back of the seat on you.

Of necessity your body has to be in a state of compression otherwise the parts your body remote from the seat back would not have a net force on them and so could not be accelerating.

You "feel" being in a state of compression and interpret this by being acted on by two external forces, one of which is in the opposite direction to that of the acceleration, an external force which does not actually exist.

If you are hanging on to the end of a rope and moving along a circular path the rope exerts an external on you which produces your centripetal acceleration. You "feel" the rope pulling on you but at the same time your body is stretched and it is as though there is another external force acting on you in a direction opposite to the centripetal acceleration and the force causing it.

A lift accelerating upwards will produce a compression of your body as though there is an external force pushing down on you as well as the upward external force on you due to the floor of the lift.

Here is another example.

Suppose that you are standing on the frictionless floor of a stationary train and the train then starts to accelerate in a straight line due to a force exerted by the engine on the rails.

What do you see? You see the train accelerating due to the force on it due to the rails. You are not moving relative to the rails because there is no horizontal force acting on you.

What do the seated passengers in the train observe? A train which is not accelerating relative to them and you accelerating relative to them. The passengerz know that there is a force on the carriage and they also know that there is no horizontal force on you.

In the reference frame of the train the passengers introduce a pseudo force such that the net force on the train is zero, hence the train is not accelerating relative to them and a pseudo force on you to account for the fact that you are accelerating relative e to them.

You might have experienced a similar sensation when looking out of a carriage window of a stationary strain at a carriage of another train which is on an adjacent platform and starting to move. You get the impression that you are accelerating relative to the other train.

Now suppose that you are are standing in a railway carriage and there is friction between the soles of your shoes and floor of a stationary railway carriage.

The carriage starts to move. What will happen to you?

A horizontal frictional force will act on the soles of your shoes in the direction of the acceleration of the train. That frictional force will produce a linear acceleration of your centre of mass in the direction of the acceleration of the train and also a torque about your centre of mass which will cause you to rotate backwards.

Note that the frictional force due to the floor of the carriage is the only horizontal force which is acting on you and it is a real force. It has a Newton third law pair the force that the soles of your shoes exert on the floor of the carriage.

Now what "sensation" do you "feel"? You have the sensation that there is something pushing you in the opposite direction to that of the acceleration of the train.

• So even though your net acceleration is 0 you can still feel the forces pushing on you? Commented Sep 19, 2019 at 14:21
• @Melvin In the frame of the train there is a pseudo force but it acts through your centre of mass not the soles of your feet. Commented Sep 19, 2019 at 15:49
• so you “feel” the pseudo force as a force that’s pulling you backwards? I didn’t really understand what you meant in your comment? Commented Sep 19, 2019 at 18:49
• @Melvin You “feel” as though a force is pushing you backwards. Commented Sep 19, 2019 at 19:15

What you feel is the centrifugal force, not an acceleration. Acceleration is rate of change of velocity. If you are not moving (or starting to move) in the rotating frame, your acceleration is zero in that frame.