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.