# Questions about understanding circular motion & the forces involved

• Sho Kano

#### Sho Kano

Hey, I'm having some difficulty in understanding circular motion and it's forces. Here is the situation I'm on:
Imagine you are on a smooth turn-table with smooth shoes and the table starts spinning. Because of inertia, and because the frictional force is not nearly enough to keep you going in a circle, you'll start moving in a more/less straight line towards the rim of the table. Now let's say the table had a fence along the rim, and you eventually bump into the fence and you're forced to move in a circular path.

Is the perceived gravity the same as the centrifugal force in this case? Do you experience a centrifugal force in situation (a) where you didn't hit the fence yet and (b) at the fence?

My understanding is that the centrifugal force is purely you're tendency to move tangentially. So since there is no or little resistance to that in situation (a), you don't experience anything. In (b) however, you will experience a force radially outward (??) - Why?

It seems like centrifugal forces are similar to the backward force that you experience when you accelerate inside a car.

• Delta2
Hey, I'm having some difficulty in understanding circular motion and it's forces. Here is the situation I'm on:
Imagine you are on a smooth turn-table with smooth shoes and the table starts spinning. Because of inertia, and because the frictional force is not nearly enough to keep you going in a circle, you'll start moving in a more/less straight line towards the rim of the table. Now let's say the table had a fence along the rim, and you eventually bump into the fence and you're forced to move in a circular path.

Is the perceived gravity the same as the centrifugal force in this case? Do you experience a centrifugal force in situation (a) where you didn't hit the fence yet and (b) at the fence?
To apply Newton's laws, you have to examine the motion in a non-rotating frame of reference. If there was no force acting on the body, it would fly off in a straight line in the tangential direction (Newton's first law). So there has to be a force that keeps deflecting it toward the centre of rotation. The change in velocity is inward. So there is an inward force. It is not centrifugal (away from the centre). Rather it is centripetal (toward the centre).
My understanding is that the centrifugal force is purely you're tendency to move tangentially. So since there is no or little resistance to that in situation (a), you don't experience anything. In (b) however, you will experience a force radially outward (??) - Why?
There is no force required to move in a straight line at constant speed. The tangential speed does not change. The direction changes - always changing to deflect the motion toward the centre of rotation.

Centrifugal force is a force sensation that appears in a rotating frame of reference. It is not a Newtonian force. If you analyse the forces from an inertial, non-rotating, frame of reference, there is only an inward force.

AM

It seems like centrifugal forces are similar to the backward force that you experience when you accelerate inside a car.
Yes, these are "inertial forces", which are introduced solely to make Newton's 2nd Law work in non-inertial (accelerated) frames, and account for the object's coordinate acceleration in these frames. Inertial forces don't obey Newton's 3rd Law, and you don't "experience" them as you experience interaction forces.

Do you experience a centrifugal force in situation (a) where you didn't hit the fence yet and (b) at the fence?
Depends on what you mean by "experience". To calculate the acceleration by Newtons 2nd Law in the rotating frame you always include the inertial centrifugal force, and eventually the inertial Coriolis force.

Okay, so these inertial forces do not really exist, but what accounts for the "extra weight" I feel while being accelerated?

What I understand is: In the situation of in an accelerating car, if I put a scale between the seat and my back, a non-zero number would pop up. This is analogous to just standing on a scale right? You would have a "gravity" towards the seat. And this perceived gravity is just the normal force accelerating you. So in short, you'll always perceive gravity in the opposite direction of the net force on you?

In the case of the turn-table, what I feel at the fence is the centripetal force pushing inward. What I perceive, is a apparent gravity/compression due to that force acting on my back (?)

So in short, you'll always perceive gravity in the opposite direction of the net force on you?
Yes, you "feel" weight opposite to your proper acceleration (what an accelerometer measures), due to "real" interaction forces. Inertial forces do not cause proper acceleration, so you don't "feel" them.

Okay, so these inertial forces do not really exist, but what accounts for the "extra weight" I feel while being accelerated?
The centrifugal "force" phenomenon really exists (so long as the force providing the centripetal acceleration is not gravity)! It is just that it is not a Newtonian force. If you are being swung around on a rope tied to a pole, the rope is constantly accelerating you toward the centre and you feel that pull. However, your tangential motion keeps the distance between you and the centre of rotation from decreasing. So in your frame of reference (the non-inertial, rotating and, therefore, accelerating, frame) looking just at the rope and pole, you do not sense that your motion is changing. Yet you feel a force (a tension within your body that seems to be trying to send you outward). The reason you feel any force at all is because of the mechanical nature of the applied centripetal force. The rope may be tied to your belt, which in turn applies mechanical tensions to your waist which, through inter-cellular bonds of your body's bones and tissues, creates tensions between all the cells in your body. It is that tension within your body, caused by your inertia, that you perceive as centrifugal force.

What I understand is: In the situation of in an accelerating car, if I put a scale between the seat and my back, a non-zero number would pop up. This is analogous to just standing on a scale right?
When you are accelerated forward in a car when you step on the gas you feel that something is pushing you rearward. But, when correctly analysed from an inertial frame of reference, there is only one force on you and it is the force from the seat pushing you forward. The rearward force that you exert on the seat is not caused by something pushing you into the seat. Rather it is your inertia resisting the forward push from the seat.

You would have a "gravity" towards the seat. And this perceived gravity is just the normal force accelerating you. So in short, you'll always perceive gravity in the opposite direction of the net force on you?

In the case of the turn-table, what I feel at the fence is the centripetal force pushing inward. What I perceive, is a apparent gravity/compression due to that force acting on my back (?)
Gravity is different than all other forces. There is no "inertial force effect" when the acceleration is provided by gravity. When an astronaut orbits the Earth he/she feels no centrifugal (outward) effect - no sensation of centrifugal force. This is because the force acts directly on each atom rather then through tensions within your body. Similarly, your body will not push against a spring scale because the scale is accelerating at the same rate as you are. It is only if there is a perceptible difference between the gravitational forces from one end of your body to the other that you will feel any tension within your body.

AM

• Sho Kano
There is no "inertial force effect" when the acceleration is provided by gravity.
The acceleration by gravity is locally indistinguishable from an "inertial force effect".

When an astronaut orbits the Earth he/she feels no centrifugal (outward) effect - no sensation of centrifugal force.
You never feel the inertial centrifugal force, or any other inertial force.

Thanks guys I think I got it now. What accounts for what you feel are the forces within your body. In an accelerating car the seat pushes your back; because your chest is not initially accelerating, there will be somewhat of a compression in your body.
If you are being swung around on a rope tied to a pole, the rope is constantly accelerating you toward the centre and you feel that pull. However, your tangential motion keeps the distance between you and the centre of rotation from decreasing. So in your frame of reference (the non-inertial, rotating and, therefore, accelerating, frame) looking just at the rope and pole, you do not sense that your motion is changing. Yet you feel a force (a tension within your body that seems to be trying to send you outward).
In this situation, there is actually tension instead of compression. If we replace the guy with a spring, the tension from the rope extends the top part, but the other part is left swinging around. And so a shoe would have an apparent force on it so when it flies off, it goes tangentially by virtue of inertia.

So to sum up, the apparent force or inertial force is of direct result from the real (applied) force, and it is because of inertia that any of this happens. You don't actually feel inertia.

inertial force is of direct result from the real (applied) force
Not really. Inertial forces are an artifact of the coordinate choice. In the inertial frame there is only the real force, but no inertial force.

The acceleration by gravity is locally indistinguishable from an "inertial force effect".

You never feel the inertial centrifugal force, or any other inertial force.
I am not sure what you mean by these terms. What would you say that astronauts experience when whirling around in the centrifuge with their face feeling like it is being squished into their brains?

My point is that there is a huge difference between being whirled around in a centrifuge and being whirled around the Earth in orbit. There is a similar perceptible and a real physical difference between gravitational free-fall and an accelerating rocket or car.

AM

Last edited:
So to sum up, the apparent force or inertial force is of direct result from the real (applied) force, and it is because of inertia that any of this happens. You don't actually feel inertia.
A dog tied to a 100lb cannon ball by a neck chain will experience the effect of inertia when he sees his first squirrel.

AM

Not really. Inertial forces are an artifact of the coordinate choice. In the inertial frame there is only the real force, but no inertial force.
Sure the inertial force only exists in a non-inertial frame; I meant that what you feel is due to the real force pushing on you.
A dog tied to a 100lb cannon ball by a neck chain will experience the effect of inertia when he sees his first squirrel.
Do you mean in a way that it can't move? But you can't actually "feel" inertia right? In this case the dog feels a tension force acting opposite to it's motion- which is an effect of inertia (?)

Sure the inertial force only exists in a non-inertial frame; I meant that what you feel is due to the real force pushing on you.

Do you mean in a way that it can't move? But you can't actually "feel" inertia right? In this case the dog feels a tension force acting opposite to it's motion- which is an effect of inertia (?)
The effects are felt regardless of which frame you are doing the analysis in. If you are experiencing it but analysing it (ie. by applying Newton's laws) in an inertial frame, you conclude that there is only the Newtonian force acting on you and what you feel is the effect of your own inertia. If you are experiencing it and analysing it your non-inertial frame (ie. you assume that you are not accelerating) you will conclude that there must be a force is being applied in the opposite direction (ie. opposite to which the actual Newtonian force is being applied).

AM

The effects are felt regardless of which frame you are doing the analysis in. If you are experiencing it but analysing it (ie. by applying Newton's laws) in an inertial frame, you conclude that there is only the Newtonian force acting on you and what you feel is the effect of your own inertia. If you are experiencing it and analysing it your non-inertial frame (ie. you assume that you are not accelerating) you will conclude that there must be a force is being applied in the opposite direction (ie. opposite to which the actual Newtonian force is being applied).

AM
Right, in an inertial frame there must be only a Newtonian force acting on me, so what I feel is explained by inertia. In an non-inertial frame however, all I know is that I feel a force in the opposite direction- and that is explained by the inertial force. In both situations though, what I feel is the same sensation, and that can be explained by the compression/stretching the earlier posts. Is this right?

Right, in an inertial frame there must be only a Newtonian force acting on me, so what I feel is explained by inertia. In an non-inertial frame however, all I know is that I feel a force in the opposite direction- and that is explained by the inertial force. In both situations though, what I feel is the same sensation, and that can be explained by the compression/stretching the earlier posts. Is this right?
Basically, that's right. In the case of you rotating about a central point, there is only the Newtonian centripetal force acting on you. The force that seems to be counteracting the centripetal force (when applying Newton's laws in your rotating frame treated as if it were an inertial frame), is sometimes referred to as an inertial force but that just confuses everyone. It is an inertial effect. No force is needed to keep you out there.

AM

Sure the inertial force only exists in a non-inertial frame; I meant that what you feel is due to the real force pushing on you.
Yes. Your coordinate acceleration (dv/dt) is frame dependent, and so are inertial forces. What you "feel" is frame independent, and so are the real forces.

I am not sure what you mean by these terms. What would you say that astronauts experience when whirling around in the centrifuge with their face feeling like it is being squished into their brains?
They experience deformations due to the non-uniformly applied real centripetal force. This is a frame independent effect, while inertial forces are frame dependent.

My point is that there is a huge difference between being whirled around in a centrifuge and being whirled around the Earth in orbit.
Yes, because here (in Newtonian terms) the centripetal force is applied approximately uniformly to a small body, so it doesn't cause much deformation.

• nrqed
The acceleration by gravity is locally indistinguishable from an "inertial force effect".

You never feel the inertial centrifugal force, or any other inertial force.
Perhaps it is a matter of terminology. I use the term "inertial force effect" or "inertial effect" to refer to the perceived force (perceived by application of Newton's laws in the non-inertial frame of reference). Inside a non-rotating free falling or orbiting space-craft, there would be no perceived forces. But inside that same space-craft being hurled around in a centrifuge, forces would be perceived (by application of Newton's laws in the space-craft frame). So, in that sense, I would say that gravity is distinguishable from an "inertia force effect" - in the sense that there is no local inertial force effect in the non-inertial frame if the only real forces are gravitational.

AM

Perhaps it is a matter of terminology.
Yes, as usual you prefer a non-standard one.

Inside a non-rotating free falling or orbiting space-craft, there would be no perceived forces
Inertial forces have nothing to do with inside or outside. They depend on the chosen reference frame, which extends to infinity, and includes the inside and outside of the space-craft.

• nrqed
Yes, as usual you prefer a non-standard one.

Inertial forces have nothing to do with inside or outside. They depend on the chosen reference frame, which extends to infinity, and includes the inside and outside of the space-craft.
Ok. We are in agreement on that. But what is it that leads the observer in the reference frame of the free-falling or orbiting spacecraft to posit an outward force acting on the space-craft and its contents?

AM

Ok. We are in agreement on that. But what is it that leads the observer in the reference frame of the free-falling or orbiting spacecraft to posit an outward force acting on the space-craft and its contents?
If he is using Newtonian physics a free falling frame under gravity is non-inertial. In GR a free falling frame is inertial and there are no inertial forces acting.

If he is using Newtonian physics a free falling frame under gravity is non-inertial. In GR a free falling frame is inertial and there are no inertial forces acting.
I was trying to understand your statement: the acceleration by gravity is locally indistinguishable from an "inertial force effect". Did you mean to say "the frame of reference of a body undergoing acceleration by gravity is locally indistinguishable from an inertial reference frame"? I don't see inertial forces arising in such a frame.

AM

I don't see inertial forces arising in such a frame.
That's the GR interpretation.

That's the GR interpretation.
It is also the conclusion one reaches by applying Newton's laws of motion in the non-inertial, falling reference frame.

AM

It is also the conclusion one reaches by applying Newton's laws of motion in the non-inertial, falling reference frame.
No, it isn't. If the frame is non-inertial, then there are inertial forces.

No, it isn't. If the frame is non-inertial, then there are inertial forces.
Perhaps you could elaborate. I don't see the inertial forces in the non-inertial frame of reference of a free-falling or orbiting space-craft.

AM

I don't see the inertial forces in the non-inertial frame of reference
You can't see forces in general. Inertial forces are introduced to make Newtons 2nd Law work in non-inertial frames. This was explained in post #3 already.

You can't see forces in general. Inertial forces are introduced to make Newtons 2nd Law work in non-inertial frames. This was explained in post #3 already.
??. But that is my point. One does not need to introduce inertial forces in order to make Newton's laws work in a non-inertial frame where the acceleration is caused by gravity.

AM

Perhaps you could elaborate. I don't see the inertial forces in the non-inertial frame of reference of a free-falling or orbiting space-craft.
The space-craft is at rest in this frame, yes? In the Newtonian model, gravity is a real force in this frame, yes? The inertial force is the one holding the space-craft in place against the real force of gravity.

Edit: of course, we could adopt the GR model and consider that gravity is not a real force. But then the frame is inertial which goes against the claim that the frame is non-inertial.

• nrqed
One does not need to introduce inertial forces in order to make Newton's laws work in a non-inertial frame where the acceleration is caused by gravity.
Assuming Newton's model of Gravity (real frame independent interaction force), you have to introduce an inertial force to make Newton's 2nd Law work in a free falling frame.

The space-craft is at rest in this frame, yes? In the Newtonian model, gravity is a real force in this frame, yes? The inertial force is the one holding the space-craft in place against the real force of gravity.

Edit: of course, we could adopt the GR model and consider that gravity is not a real force. But then the frame is inertial which goes against the claim that the frame is non-inertial.

Assuming Newton's model of Gravity (real frame independent interaction force), you have to introduce an inertial force to make Newton's 2nd Law work in a free falling frame.
The only difficulty here is: how do you determine that there is a force of gravity? There is no experiment that you can do to measure the gravitational force.

If you were in a capsule being hurled around a centrifuge (unbeknownst to you), treating your non-inertial frame as inertial, you would have to postulate some force of unknown origin pulling things outward. You could measure these forces using a spring, for example.

If you were in a charged capsule being accelerated in the direction of an oppositely charged body and treating your non-inertial frame of reference as if it were inertial, you could do a local experiment and you would conclude that non-charged bodies moved differently than charged bodies so you would have to postulate mysterious forces that were like electrical forces except that they operated on non-charged bodies and acted differently, or not at all, on charged bodies.

But if the accelerating force is a locally uniform gravitational force (which you have no means of detecting if you treat your non-inertial frame as inertial) there is no need for additional forces in order to make Newton's laws of motion work in your free-falling frame of reference.

As far as GR is concerned, is this not just another way of looking at the principle of equivalence? Frames of reference accelerating in a locally uniform gravitational field do not require fictitious (inertial) forces in order to apply the laws of motion.

AM

how do you determine that there is a force of gravity?
Newton's law of universal gravitation tells you so.

there is no need for additional forces in order to make Newton's laws of motion work in your free-falling frame of reference.
If you don't use Newton's version of gravitation. That's what GR does, where the free falling frame is considered locally inertial.

• nrqed
??. But that is my point. One does not need to introduce inertial forces in order to make Newton's laws work in a non-inertial frame where the acceleration is caused by gravity.
treating your non-inertial frame as inertial
There is no experiment that you can do to measure the gravitational force.
If you have a frame that is experimentally indistinguishable from inertial, why are you calling it non-inertial?

Newton's law of universal gravitation tells you so.
How can the observer in the free-falling frame apply Newton's law of universal gravitation if he can't detect gravity?

AM

How can the observer in the free-falling frame apply Newton's law of universal gravitation if he can't detect gravity?
If the frame is "non-inertial" due to gravity then you have stipulated that gravity exists whether it is detected or not. There's no sense quibbling about what you've already stipulated to.

In any case, a free falling observer can easily detect gravity. He looks around at the other free falling objects. [This does not count against the equivalence principle since such observations are non-local]