The gyroscope and its ability to avoid "falling down"

[Luigi Fortunati]

The feature of the gyroscope is its ability to prevent the falling down of an object that rotates.

Only a force directed upward can oppose a downward force.

In the case of the spinning slingshot, we see that the stone tends to go up when the angular velocity increases.

To identify the force that pushes the stone upward we must identify the 3 forces that act on it:
- gravity (constant and vertically directed downwards)
- the force FC=mw^2r centripetal which has the same inclined direction of the string and where r is exactly the length of the rope
- the force FF=mw^2r' centrifuge that goes outwards in a horizontal direction and where r' is the radius of the circumference traveled by the stone.

Thus the centripetal and centrifugal forces have different directions and modules, so they do not cancel and their result (at constant angular velocity) is directed towards the HIGH with the same module of gravity.

This force that push up the stone of the spinning sling is the same that prevents the spinning top from falling down.

This is the force that opposes gravity to generate the gyroscope effect.

 

[A.T.]

The feature of the gyroscope is its ability to prevent the falling down of an object that rotates.

No, it doesn't prevent it from falling down. A spinning gyroscope falls down just like a non-spinning does. The rotation just prevents the axis to tilt, like it would for a non-rotating one for the same torque.

This force that push up the stone of the spinning sling is the same that prevents the spinning top from falling down.

It is trivial that the support force (attachment) hast to balance the weight of an object.

 

[Drakkith]

To identify the force that pushes the stone upward we must identify the 3 forces that act on it:
- gravity (constant and vertically directed downwards)
- the force FC=mw^2r centripetal which has the same inclined direction of the string and where r is exactly the length of the rope
- the force FF=mw^2r' centrifuge that goes outwards in a horizontal direction and where r' is the radius of the circumference traveled by the stone.

Centrifugal force is a fictitious force, so it can't act on the stone unless you want to complicate things by taking a rotating reference frame. Otherwise, there are only two forces acting on the stone, the force provided by the string (part of which is the centripetal force) and the force of gravity. Which makes sense given that the stone is constantly accelerating, which requires a net force.

Additionally, the centripetal force doesn't act along the string, but is just the horizontal component of the force provided by the string. Imagine the circular path that the stone takes and you'll see that the radius of the circle does not start at the origin of the string, but somewhere below it. If you try to use your formula using the length of the string as your radius, you'll get an incorrect answer.

Thus the centripetal and centrifugal forces have different directions and modules, so they do not cancel and their result (at constant angular velocity) is directed towards the HIGH with the same module of gravity.

The force that causes the stone to rise is simply the vertical component of the force provided by the string. At a constant angular velocity the horizontal component of the string's force keeps the stone moving in a circular path while the vertical component keeps it from falling.

Also, the centrifugal and centripetal forces always point directly opposite of each other, so there's no way that their sum can be directed anywhere since they cancel out.

 

[Luigi Fortunati]

Is this "trivial" force (or not) exactly the result of the two centrifugal and centripetal forces acting on the stone?

 

[Drakkith]

Is this "trivial" force (or not) exactly the result of the two centrifugal and centripetal forces acting on the stone?

No, those two forces cancel out completely as I explained above.

 

[Luigi Fortunati]

Centrifugal force is a fictitious force, so it can't act on the stone unless you want to complicate things by taking a rotating reference frame. Otherwise, there are only two forces acting on the stone, the force provided by the string (part of which is the centripetal force) and the force of gravity. Which makes sense given that the stone is constantly accelerating, which requires a net force.

Additionally, the centripetal force doesn't act along the string, but is just the horizontal component of the force provided by the string. Imagine the circular path that the stone takes and you'll see that the radius of the circle does not start at the origin of the string, but somewhere below it. If you try to use your formula using the length of the string as your radius, you'll get an incorrect answer.
The force that causes the stone to rise is simply the vertical component of the force provided by the string. At a constant angular velocity the horizontal component of the string's force keeps the stone moving in a circular path while the vertical component keeps it from falling.

Also, the centrifugal and centripetal forces always point directly opposite of each other, so there's no way that their sum can be directed anywhere since they cancel out.

The force that the rope exerts on the stone is directed towards the hand, and not in other directions.

 

[Drakkith]

The force that the rope exerts on the stone is directed towards the hand, and not in other directions.

And that force can be broken down into horizontal and vertical components. This is basic mechanics stuff. You can find it in any physics textbook and many places online.

 

[Luigi Fortunati]

No, those two forces cancel out completely as I explained above.

No, those two forces cancel out completely as I explained above.

The force that the rope exerts on the stone is directed towards the hand, and not in other directions: with what forces do you cancel?

 

[Drakkith]

The force that the rope exerts on the stone is directed towards the hand, and not in other directions: with what forces do you cancel?

Look, we're not going to make any progress in our discussion if you don't understand how a force can be broken down into different components. I highly recommend reading up on that before trying to understand gyroscopes and slings.

 

[Luigi Fortunati]

And that force can be broken down into horizontal and vertical components. This is basic mechanics stuff. You can find it in any physics textbook and many places online.

Ok, let's break it down: one force is vertical and cancels gravity, the other is horizontal and cancels the centrifugal force.

 

[Luigi Fortunati]

Look, we're not going to make any progress in our discussion if you don't understand how a force can be broken down into different components. I highly recommend reading up on that before trying to understand gyroscopes and slings.

Look, we're not going to make any progress in our discussion if you don't understand me.

 

[Drakkith]

Ok, let's break it down: one force is vertical and cancels gravity, the other is horizontal and cancels the centrifugal force.

The force exerted by the string can be broken down into horizontal and vertical components. The horizontal component is the centripetal force that keeps the stone moving in a circle. This force accelerates the stone is not canceled out by anything unless you want to move to a rotating frame and invoke centrifugal force. In such a case the centripetal force cancels out the centrifugal force. If it didn't, the stone would move towards or away from the center of rotation.

The vertical component of the rope's force counteracts gravity.

 

[Luigi Fortunati]

The force exerted by the string can be broken down into horizontal and vertical components. The horizontal component is the centripetal force that keeps the stone moving in a circle. This force accelerates the stone is not canceled out by anything unless you want to move to a rotating frame and invoke centrifugal force. In such a case the centripetal force cancels out the centrifugal force. If it didn't, the stone would move towards or away from the center of rotation.

The vertical component of the rope's force counteracts gravity.

And what happens if the angular velocity increases? It happens that the strength of the rope increases and breaks down into two forces that exceed the opposite ones.
Increases the upward force that exceeds gravity and pushes the stone up.
And the horizontal component also increases.

 

[Dale]

- the force FC=mw^2r centripetal which has the same inclined direction of the string and where r is exactly the length of the rope

This is all wrong. The centripetal force is only the horizontal component of the tension. The tension is greater than the centripetal force. The centripetal force is exactly horizontal and the tension is in the inclined direction

This force that push up the stone of the spinning sling is the same that prevents the spinning top from falling down.

So you claim the vertical component of the tension in the rope prevents the spinning top from falling down? There is no rope in the gyroscope.

Analogies are great, but they break down at some point. You have pushed this one past the breaking point.

 

[Drakkith]

And what happens if the angular velocity increases? It happens that the strength of the rope increases and breaks down into two forces that exceed the opposite ones.

While the angular velocity is increasing, the upward component exceeds gravity, causing the stone to rise. The horizontal component increases as well in order to keep the stone moving in a circle at the increased angular velocity. Once you reach a steady angular velocity, the horizontal component falls until it cancels the force of gravity, just as it had before. However the horizontal component maintains its increased magnitude since the stone is now moving faster and requires a stronger force to keep it moving in a circle.

 

[Luigi Fortunati]

While the angular velocity is increasing, the upward component exceeds gravity, causing the stone to rise. The horizontal component increases as well in order to keep the stone moving in a circle at the increased angular velocity. Once you reach a steady angular velocity, the horizontal component falls until it cancels the force of gravity, just as it had before. However the horizontal component maintains its increased magnitude since the stone is now moving faster and requires a stronger force to keep it moving in a circle.

Exactly! As the angular velocity increases, the component increases upward and this causes the stone to rise and cause the slope of the rope to decrease.
Then a new equilibrium is achieved when the decreased inclination reduces the upward thrust until it becomes the same as the opposite gravity force.

 

[A.T.]

Look, we're not going to make any progress in our discussion if you don't understand me.

Your analogy makes no sense, because in your example there is no re-orienting of the plane of rotation, which is the whole point of the gyroscopic effect.

 

[Luigi Fortunati]

This is all wrong. The centripetal force is only the horizontal component of the tension. The tension is greater than the centripetal force. The centripetal force is exactly horizontal and the tension is in the inclined direction

So you claim the vertical component of the tension in the rope prevents the spinning top from falling down? There is no rope in the gyroscope.

Analogies are great, but they break down at some point. You have pushed this one past the breaking point.

I called centripetal force the force that goes from the stone to the hand, you call it the force of tension of the rope, it changes only the name that we are leaning on and nothing else.

I confirm that the mechanism is the same and I will show it after having clarified the question of the sling.

 

[A.T.]

I will show it after having clarified the question of the sling.

If you are looking for an analogy to the gyroscopic procession based on a single mass in circular motion, then see the video below.

 

[Dale]

I called centripetal force the force that goes from the stone to the hand

This is wrong.

The centripetal force is *defined* as the force which points towards the center of the circular path. The force that goes from the stone to the hand (the tension) does not point in that direction, it points above the center. It therefore cannot be the centripetal force, and your labeling it thus is wrong. You are not free to redefine standard terms like "centripetal force".

 

[Luigi Fortunati]

This is wrong.

The centripetal force is *defined* as the force which points towards the center of the circular path. The force that goes from the stone to the hand (the tension) does not point in that direction, it points above the center. It therefore cannot be the centripetal force, and your labeling it thus is wrong. You are not free to redefine standard terms like "centripetal force".

Ok, I do not call it centrifugal force, I call it "rope tension".

This force has a vertical component that goes up and counteracts gravity, ok?

When the speed of rotation increases, this "tension of the rope" increases and also the component (which goes upwards) increases and exceeds the force of gravity, therefore "pulls" the stone upwards.

Is that okay?

 

[Dale]

Ok, I do not call it centrifugal force, I call it "rope tension".

This force has a vertical component that goes up and counteracts gravity, ok?

Yes.

When the speed of rotation increases, this "tension of the rope" increases and also the component (which goes upwards) increases and exceeds the force of gravity, therefore "pulls" the stone upwards.

Is that okay?

Yes, any upward acceleration of the center of mass is due to the vertical component of the tension.

 

[CWatters]

To identify the force that pushes the stone upward we must identify the 3 forces that act on it:
- gravity (constant and vertically directed downwards)
- the force FC=mw^2r centripetal which has the same inclined direction of the string and where r is exactly the length of the rope
- the force FF=mw^2r' centrifuge that goes outwards in a horizontal direction and where r' is the radius of the circumference traveled by the stone.

I think others have already pointed this out but, even if we correct Luigi's definition of centripetal force (to the horizontal component of the tension in the rope) he is still wrong.

If there were three forces and they "balance out" ( aka vector sum to zero in the horizontal plane) then the stone should move in a straight line. In order to move in a circle there must be a net force acting on the stone.

 

[Luigi Fortunati]

Yes.

Yes, any upward acceleration of the center of mass is due to the vertical component of the tension.

Ok.

The force of the string tension is equal to the centripetal (or centrifugal) force FC divided sine_of_alpha (where alpha is the angle between the string and the vertical)?

 

[Luigi Fortunati]

I think others have already pointed this out but, even if we correct Luigi's definition of centripetal force (to the horizontal component of the tension in the rope) he is still wrong.

If there were three forces and they "balance out" ( aka vector sum to zero in the horizontal plane) then the stone should move in a straight line. In order to move in a circle there must be a net force acting on the stone.

You're right, this tell us the school books.

They tell us that only the centripetal force acts on the stone, not the centrifugal force which is "apparent".

If so, the only net force is the centripetal force.

But then, if there is no real centrifugal force, who pushes the stone outwards when the angular velocity increases?

 

[jbriggs444]

But then, if there is no real centrifugal force, who pushes the stone outwards when the angular velocity increases?

Nobody. Fictitious forces have no third law partners.

Viewed from the inertial frame there an inward force and an inward acceleration. There is no outward force.

Viewed from the rotating frame there is an inward force and no inward acceleration. We invent an outward "centrifugal" force to explain the lack of inward acceleration.

Edit: If you want to pick and choose which rotating frame to use, you may find that Coriolis and Euler forces are involved. Nobody exerts those either.

 

[Drakkith]

But then, if there is no real centrifugal force, who pushes the stone outwards when the angular velocity increases?

The force doing the acceleration of the angular velocity, or one of its components. For a sling, this is the tension in the rope. To accelerate the stone, the person must move their hand around so that the tension is continuously pointing slightly ahead of the stone. This provides a net force in the direction tangential to the stones path and thus accelerates it.

 

[Dale]

The force of the string tension is equal to the centripetal (or centrifugal) force FC divided sine_of_alpha (where alpha is the angle between the string and the vertical)?

Yes, for the centripetal force. There is no centrifugal force in this scenario.

If you want to introduce a centrifugal force then you need to specify the reference frame.

But then, if there is no real centrifugal force, who pushes the stone outwards when the angular velocity increases?

The stone is never pushed outwards. It’s acceleration is at all times approximately towards the center. It never accelerates outwards.

 

[Luigi Fortunati]

Yes, for the centripetal force. There is no centrifugal force in this scenario.

If you want to introduce a centrifugal force then you need to specify the reference frame.

The stone is never pushed outwards. It’s acceleration is at all times approximately towards the center. It never accelerates outwards.

It never accelerates outwards?!?

When the angular velocity increases, the stone goes to cover a larger circumference, then accelerates!

 

[Luigi Fortunati]

The force doing the acceleration of the angular velocity, or one of its components. For a sling, this is the tension in the rope. To accelerate the stone, the person must move their hand around so that the tension is continuously pointing slightly ahead of the stone. This provides a net force in the direction tangential to the stones path and thus accelerates it.

The tension of the rope accelerates the stone, ok, but the stone is not only faster, the stone also accelerates outwards, towards the centrifugal direction!

The rope "pulls" towards the center, not towards the outside!

 

[Drakkith]

It never accelerates outwards?!?

When the angular velocity increases, the stone goes to cover a larger circumference, then accelerates!

The tension of the rope accelerates the stone, ok, but the stone is not only faster, the stone also accelerates outwards, towards the centrifugal direction!

Not quite. The stone's radius increases, yes, but this is not because it accelerates outwards. This is because the inward pull from the rope is not sufficient to keep it on a circular path. As the stone moves outwards, the inclination of the rope decreases and more of the tension is exerted horizontally until the centripetal force becomes sufficient to keep it moving in a circle.

The rope "pulls" towards the center, not towards the outside!

Inertia "pulls" it outwards. I have the word pull in quotes because nothing is actually pulling (or pushing) on the stone in the outward radial direction.

 

[jbriggs444]

The tension of the rope accelerates the stone, ok, but the stone is not only faster, the stone also accelerates outwards, towards the centrifugal direction!

The stone never accelerates outward. Its outward movement is possible because "outward" is not a fixed direction. An inward acceleration now results in a velocity increment in a direction that is outward one half revolution from now.

Edit: And, more to the point, a forward tangential acceleration now results in a velocity increment in a direction that is outward a quarter revolution from now.

 

[Dale]

It never accelerates outwards?!?

Correct.

When the angular velocity increases, the stone goes to cover a larger circumference, then accelerates!

And during that motion it never at any time accelerates outwards. The acceleration remains approximately inwards at all times. I encourage you to work through the math on this.

the stone also accelerates outwards, towards the centrifugal direction!

No, it does not. At no time is the acceleration ever outwards.

 

[Luigi Fortunati]

Not quite. The stone's radius increases, yes, but this is not because it accelerates outwards. And for what, then?

This is because the inward pull from the rope is not sufficient to keep it on a circular path.

If the pull of the rope inwards is not sufficient, it means that it is less than the opposite force, the one that goes outwards.

As the stone moves outwards, the inclination of the rope decreases and more of the tension is exerted horizontally until the centripetal force becomes sufficient to keep it moving in a circle.

Exact.

And why does the centripetal force have to increase? To counteract the increase in the opposing force.

Inertia "pulls" it outwards. I have the word pull in quotes because nothing is actually pulling (or pushing) on the stone in the outward radial direction.

It is certainly true that the inertia of the stone pulls outwards.

But what is this thing that we call "inertia"? It is the property of the mass not to exert force if it does not undergo it (first principle) but to react with real force when it is subjected (according to the principle F = ma: the accelerated mass reacts with the force F).

The inertia of the stone that "pulls" towards the outside is a force that is anything but "apparent", because it is a mass that moves with *accelerated* moviment.

 

[Luigi Fortunati]

Correct.

And during that motion it never at any time accelerates outwards. The acceleration remains approximately inwards at all times. I encourage you to work through the math on this.

No, it does not. At no time is the acceleration ever outwards.

When the angular velocity increases, the stone goes to cover a larger circumference outside the smaller one.

No bigger circumference can stay inside a smaller one!

Geometrically.

 

[Drakkith]

It is certainly true that the inertia of the stone pulls outwards.

But what is this thing that we call "inertia"? It is the property of the mass not to exert force if it does not undergo it (first principle) but to react with real force when it is subjected (according to the principle F = ma: the accelerated mass reacts with the force F).

The inertia of the stone that "pulls" towards the outside is a force that is anything but "apparent", because it is a mass that moves with *accelerated* moviment.

The stone obviously exerts a force on the rope that points directly opposite of the tension. But we're not talking about the force on the rope, we're talking about the force on, and motion of, the stone. And the only acceleration going on here is the stone's acceleration towards the center or, if in the process of increasing its angular velocity, its acceleration tangentially to its motion. Not outwards.

When the angular velocity increases, the stone goes to cover a larger circumference outside the smaller one.

No bigger circumference can stay inside a smaller one!

Geometrically.

Of course. No one is arguing otherwise. But this movement isn't an acceleration outwards. No force is ever exerted on the stone in the outward radial direction so no acceleration can take place.

Before replying, please do the following: draw a picture of the stone and the rope. Label the real forces (not fictitious) on the stone. You should only have 2 forces, neither of which are capable of accelerating the stone outwards.

 

[Dale]

When the angular velocity increases, the stone goes to cover a larger circumference outside the smaller one.

No bigger circumference can stay inside a smaller one!

Geometrically.

Yes, clearly.

You appear to be confused about the distinction between velocity and acceleration. The velocity indeed has an outward component, but the acceleration only has an inward component.

Please work out the math for yourself to convince yourself of this fact: Take a spiral path and calculate the first derivative (velocity) and the second derivative (acceleration). What is the sign of the radial component of each?

If you get stuck, post your work and we can help.

 

[CWatters]

When the angular velocity increases, the stone goes to cover a larger circumference outside the smaller one.

No bigger circumference can stay inside a smaller one!

Geometrically.

Earlier you were talking about the angular velocity increasing and we were answering accordingly.

Now you seem to be talking about the rope getting longer. That's the only way the circumference can increase.

 

[Dale]

Earlier you were talking about the angular velocity increasing and we were answering accordingly.

Now you seem to be talking about the rope getting longer. That's the only way the circumference can increase.

No, he is correct on this point. Suppose that you have a rope of 50 cm length. If you spin very slowly then you might have a 30 cm radius with the rope hanging down 40 cm. If you spin a little faster then the rope will hang down only 30 cm and the radius will be 40 cm. In both cases the rope is 50 cm.

 

[Luigi Fortunati]

Earlier you were talking about the angular velocity increasing and we were answering accordingly.

Now you seem to be talking about the rope getting longer. That's the only way the circumference can increase.

I have never talked about the lengthening of the rope but the lengthening of the circumference traveled by the stone as the angular velocity increases.

 

[CWatters]

Ok I meant increasing the radius.

 

[Luigi Fortunati]

The stone obviously exerts a force on the rope that points directly opposite of the tension. But we're not talking about the force on the rope, we're talking about the force on, and motion of, the stone.

The two things are absolutely connected to each other: it is obvious that if the stone exerts a force on the rope, this force must have it.

The stone pulls this force out of its inertia which is "inert" only if no one stimulates it.

If the rope (with real force) pulls the stone to one side, it is obvious that the stone responds by pulling the other side with its own inertia that stops being "inert" and becomes "real" force!

And the only acceleration going on here is the stone's acceleration towards the center or, if in the process of increasing its angular velocity, its acceleration tangentially to its motion. Not outwards.

The tangential velocity is directed towards the outside even if not radially.

And it is precisely the increase of this speed that makes the circumference of the stone path widen.

Of course. No one is arguing otherwise. But this movement isn't an acceleration outwards. No force is ever exerted on the stone in the outward radial direction so no acceleration can take place..

The force that radially accelerates the stone outwards is that of its own inertia which is activated when the rope pulls inward.

It is exactly the same and opposite force provided by the third principle which arises spontaneously when the horizontal component of the tension of the rope acts on the stone.

Before replying, please do the following: draw a picture of the stone and the rope. Label the real forces (not fictitious) on the stone. You should only have 2 forces, neither of which are capable of accelerating the stone outwards.

This is precisely what should make you think!

If the forces were only 2 the stone could not accelerate outwards and could not extend its rotation radius!

The forces are 3 (can not be 2!).

 

[Dale]

The force that radially accelerates the stone outwards is that of its own inertia which is activated when the rope pulls inward.

I am getting tired of repeating this. Please do the math. The stone does not accelerate outwards. At all times the acceleration is inwards, so there is no force outwards.

The tangential velocity is directed towards the outside even if not radially.

Velocity, yes. Acceleration, no.

If the forces were only 2 the stone could not accelerate outwards

It doesn’t accelerate outwards.

and could not extend its rotation radius!

This does not follow. Even while it extends the rotation radius, the acceleration is still inwards. Do the math.

The forces are 3 (can not be 2!).

There are only two forces acting on the stone: tension and gravity.

 

[Drakkith]

@Luigi Fortunati I'm done. It's pretty obvious you have no wish to actually learn what's going on here, so I'll just bow out of this thread. Have a nice day.

 

[A.T.]

The forces are 3 (can not be 2!).

In the inertial frame there are 2 forces. In some non-inertial frame might be more.

If the forces were only 2 the stone could not accelerate outwards...

It doesn't accelerate outwards in the inertial frame.

...and could not extend its rotation radius!

It can extend it's radius by accelerating inwards less than required for a constant radius.

 

[rcgldr]

Maybe this will help, at the connection between rope and stone, in a steady state, the rope exerts a tension force on the stone, and the stone exerts an equal and opposite force on the rope. The rope's horizontal force on the stone is a centripetal force, the stones horizontal force on the rope is an outwards reaction force, sometimes called a centrifugal reaction force (not to be confused with the fictitious centrifugal force of a rotating frame of reference). The rope also exerts an upwards vertical force onto the stone, and the stone exerts a downwards vertical force on the string. Gravity exerts a downwards force on the stone, which opposes the rope's upwards vertical force on the stone, and in a steady state, the net vertical force on the stone is zero, so the only net force on the stone is a centripetal force causing the stone to follow a circular path (again assuming a steady state).

 

[rcgldr]

I assume the gyroscope issue is how a gyroscope can precess and remain horizontal (or nearly so) while only supported at one end of the gyroscope. In this case, the support exerts an upwards force onto the end of the gyroscope, and the gyroscope exerts a downwards force onto the support. Gravity exerts a downwards force on the gyroscope that opposes the upwards force that the support exerts on the gyroscope so it has zero net vertical force (in an ideal state). The upwards force at the support and the downwards force at the gyroscope's center of mass produces a torque that would tend to lower the gyroscope. The precession generates a torque that would tend to raise the gyroscope. In an ideal state, the torques cancel and the gyroscope remains horizontal while precessing.

If the support is a string suspended from above, then ideally, the gyroscopes center of mass moves about a horizontal plane as the support point of the string moves inwards and outwards in a spiraling pattern. For an example of this, go to video #9 (top link is a .wmv file) on this web page. Note that the professor starts the gyro tilted upwards a bit to induce the spiral like pattern, and later he's not quite happy when the gryo support point goes through one outwards and inwards pattern, then tends to stay outwards for a while and he interferes with it. I'm wondering if the support path pattern is really supposed to cycle or instead tend to approach some constant radius circular path. I'm not sure of the dampening factors or the issue of the gyroscope slowing down over time.

http://www.gyroscopes.org/1974lecture.asp

 

[CWatters]

The two things are absolutely connected to each other: it is obvious that if the stone exerts a force on the rope, this force must have it.

The stone pulls this force out of its inertia which is "inert" only if no one stimulates it.

If the rope (with real force) pulls the stone to one side, it is obvious that the stone responds by pulling the other side with its own inertia that stops being "inert" and becomes "real" force!

If the centripetal force and the centrifugal force are both "real" and equal magnitude then their vector sum would be zero (because they act in opposite directions) and the stone would travel in a straight line. If the stone moves in a circle there must be a net force acting on it towards the centre.

 

[Luigi Fortunati]

If the centripetal force and the centrifugal force are both "real" and equal magnitude then their vector sum would be zero (because they act in opposite directions) and the stone would travel in a straight line. If the stone moves in a circle there must be a net force acting on it towards the centre.

If on the stone there was a net force that acts towards the center, the stone would move towards the center.

If on the stone there was a net force directed towards the outside, the stone would move away from the center.

Instead the stone does not approach and does not move away from the center of rotation, a clear sign that the two opposing forces (centripetal and centrifugal) cancel each other out.

Their vector sum is actually equal to zero.

But then, why is the motion not straight in the inertial reference?

Because the two forces (centripetal and centrifugal) also rotate and cancel each other at different points in the circumference.

And in fact, even in the inertial reference, the stone does not approach and does not move away from the center of rotation.

 

[weirdoguy]

If on the stone there was a net force that acts towards the center, the stone would move towards the center.

No it wouldn't because it's velocity is perpendicular to net force. Direction of the net force alone does not tell you how the stone will move. You need "initial conditions", and in this case, the initial velocity is the most importnat (it's direction).

Their vector sum is actually equal to zero. But then, why is the motion not straight in the inertial reference?

Because the net force is not zero. You should really start reading what other people write to you, because otherwise this discussion is pointless. You have some BIG misunderstandings about inertial forces and Newton's second law and you really don't want to give up on them.