Torque in a non inertial reference frame

[McLaren Rulez]

Hi,

This is partly inspired by the questions in the thread about normal forces on a cornering car but I thought I'd post here instead of mix that thread up with my question. So suppose we have a biker leaning into a corner. There are the normal force and the force of friction which act at the contact point between the wheel and the road. The weight of the biker acts downward from the CG.

Let's consider the contact between the wheel and the road as our axis of rotation. If I look at it from the non inertial frame of the bike, there are two torques: The weight and the centrifugal force and they are in opposite directions so they can balance each other out and the biker's lean angle remains constant.

However, if we look at this from an inertial frame, there is only one torque: The torque which arises due to the weight and this should make the biker lean even more (and eventually fall). There is nothing to compensate this torque so what am I missing here? How should I correctly analyze this from an inertial frame?

Thank you!

 

[the.drizzle]

Because the cyclist's wheels are rotating about an axis perpendicular (or at least non-parallel) to the axis that defines the corner the cyclist is going around.; you're missing the fact that there is rotational motion present.

If the cyclist were to somehow manage this corner with stationary wheels, then they would indeed have problems staying upright...

 

[kmarinas86]

Hi,

This is partly inspired by the questions in the thread about normal forces on a cornering car but I thought I'd post here instead of mix that thread up with my question. So suppose we have a biker leaning into a corner. There are the normal force and the force of friction which act at the contact point between the wheel and the road. The weight of the biker acts downward from the CG.

Let's consider the contact between the wheel and the road as our axis of rotation. If I look at it from the non inertial frame of the bike, there are two torques: The weight and the centrifugal force and they are in opposite directions so they can balance each other out and the biker's lean angle remains constant.

However, if we look at this from an inertial frame, there is only one torque: The torque which arises due to the weight and this should make the biker lean even more (and eventually fall). There is nothing to compensate this torque so what am I missing here? How should I correctly analyze this from an inertial frame?

Thank you!

If you include four additional force vectors for the ground, then you can see why.

In both cases (inertial and non-inertial), the vertical vectors should match.

In the inertial case however, the difference between the horizontal friction force on the bike (by the ground) and the centripetal force on the bike is equal and opposite to the difference between the horizontal friction force on the ground (by the bike) and the reactive centrifugal force on the ground. In other words, in the inertial case, we have not a centrifugal force on the bike, but a centripetal one created by the ground acting on the bike toward the axis of the turn. You will have to consider 8 vectors and not just 4.

 

[kmarinas86]

Because the cyclist's wheels are rotating about an axis perpendicular to the axis that defines the corner the cyclist is going around.; you're missing the fact that there is rotational motion present.

If the cyclist were to somehow manage this corner with stationary wheels, then they would indeed have problems staying upright...

I've never been satisfied with mystical explanations like this one. Also, the reality is that you do not even need wheels. Ice skates would work too. So your explanation is not only mystical, it is incorrect. The correct, non-mystical explanation is found in my previous post.

 

[kmarinas86]

By the way "gyroscopic stability" is simply the result of the fact that when you apply a force at a right angle to a body, the velocity of masses try to align with that force. The reason is obvious with force that is applied in the same direction for a long enough time; it will cause a change in velocity aligned with that force. By doing this, subsequent changes in displacement tend to align with that sustained force. This requires work to be done.

In the case for unconstant forces, if the force on mass points on the gyroscope was maintained at right angles to the motion of the mass points, the body or field which is acting on each mass point must be positioned such that the line between that body or field and each mass point is aligned with the forces between them, so as to obey Newton's laws of motion. This requires work to be done. In this case, it is not done onto the gyroscope. Instead it is done on the body acting on the gyroscope in order to obtain the proper position for generating forces on the mass points at a right angle.

Therefore, the reason why gyroscopes (and other gyroscope-like devices) can be "gyro-stabilized" is not some mystical property of precession and rotating reference frames. It has to do instead with the conservation of energy. A convincing reasoning which demonstrates this fact is the existence of counter-rotating gyros, which do in fact have the ability to stabilize objects, despite the fact that their precession effects cancel. A notable example of a technology which has variants employing counter-rotating gyros is called the gyro monorail.

 

[the.drizzle]

I've never been satisfied with mystical explanations like this one. Also, the reality is that you do not even need wheels. Ice skates would work too. So your explanation is not only mystical, it is incorrect. The correct, non-mystical explanation is found in my previous post.

Your ice skater example is actually something slightly different, and not a fair comparison I'm afraid, as skates work by locally compressing the ice to induce a (local) phase change at the metal-ice interface, inserting a liquid lubricating layer in between the two. Cornering on ice skates allows the skater to essentially build a banked surface at the interface, and this is why sharp corners require more complicated techniques, like "stepping" into the corner.

And I'm not being mystical, thanks, just brief!

 

[kmarinas86]

Your ice skater example is actually something slightly different, and not a fair comparison I'm afraid, as skates work by locally compressing the ice to induce a (local) phase change at the metal-ice interface, inserting a liquid lubricating layer in between the two. Cornering on ice skates allows the skater to essentially build a banked surface at the interface, and this is why sharp corners require more complicated techniques, like "stepping" into the corner.

And I'm not being mystical, thanks, just brief!

The phase change is nothing more then the effect of storing pressure energy. You can have rubber on concrete or you can have skates on ice. The kind of phase change has nothing to do with determining centrifugal and centripetal forces involved, aside from matters of being able to store this pressure energy in the first place. The ice skates obviously rely on friction for turning, it just so happens that the friction of the ice skate depends on how it moves over the ground. Along the length of the blade there is very little friction, but move it sideways and it works like a razor blade pressing the layer of water between the blade and the ice into the atomic lattice of the ice.

 

[kmarinas86]

If the cyclist were to somehow manage this corner with stationary wheels, then they would indeed have problems staying upright...

True, but it is not because of no net angular momentum:



A skilled rider can easily bank on a turn with this contraption.

The reason why it stays upright is because the motion along the turn provides a centripetal force on the bike by the ground and a centrifugal force of the bike due to the tendency of objects to maintain their trajectory. Together, these forces on the bike can provide the torque needed to keep bike from tipping over, provided it is sufficient.

In a famous physics example:



A wheel on a string behaves like this for three reasons:
1) In addition to the vertical tension on the string which counters the wheel's weight (though not the rotating force due to it), there is also a lateral tension on the bicycle's wheel by the string, establishing a centripetal force on the wheel, which generates a contribution to torque on the wheel's axis towards the "equator" around the string.
2) The spinning of the wheel enhances the centrifugal forces, providing an additional contribution to the lateral forces, by increasing the root mean square velocity of the bike mass along the circumference of the path taken by the wheel around the string. This also generates a contribution to torque on the wheel's axis towards the "equator" around the string.
3) The bicycle wheel must move around in a certain direction due to Coriolis forces acting on the wheel due to the interaction of the gravity with the momentum of the wheel.

The stability of the wheel of the above experiment has nothing to do establishing the precession itself. All establishing the precession does is just change the direction (not magnitude) of the angular momentum at the wheel axle, which is due to the Coriolis effect. The reason why it doesn't cause the stability is this: The torque on the wheel due to gravity and the vertical tension of the string is capable of causing angular acceleration, if not countered. The "precessional torque" does not involve an angular acceleration but a change in spin axis orientation. The derivative of a constant magnitude angular momentum with respect to time (i.e. the torque) can be non-zero if the axis is precessing (changing orientation). However, such a torque does not cause the wheel to accelerate around the axis, nor can it do anything to resist the angular acceleration that may occur on the wheel. I repeat again: It is the extra lateral force which may provide the torque to counter gravity!

 

[McLaren Rulez]

Thank you for the replies but I still can't see it.

First the ice skates argument. As far as I can tell, it doesn't matter that my wheels are spinning. the.drizzle could you explain to me once again why the fact that wheels are turning matters here? Because if I look at the thing from the non-inertial frame (i.e. the frame attached to the bike), there is a torque due to the centrifugal force $mv^{2}/r*h$ where h is the height of the CG. This is balanced by the torque due to the weight. Now, I didn't need to think about spinning wheels in this frame so I can't see why its relevant in an inertial frame.

kmarinas, the reactive centrifugal force seems to be the force exerted by the bike on the road. This is also at the contact point between the wheels and the road and hence has no torque as our axis is at the same place.

Another thing that I am confused about is that you are talking about two forces acting on the bike: A centripetal force and a horizontal friction force. But these are the same force. The friction is what gives the bike a centripetal force. As far as I can tell, there are only three forces on the bike: The weight, the normal force and the friction.

Thank you both for your explanations, I do appreciate your help very much.

 

[kmarinas86]

kmarinas, the reactive centrifugal force seems to be the force exerted by the bike on the road. This is also at the contact point between the wheels and the road and hence has no torque as our axis is at the same place.

When the bike turns, the ground is pushed away from the axis of the corner, while the bike is pushed towards the axis of the corner. Think about what happens when someone slips over a banana. If nothing is there to hold the banana in place, the banana moves in the direction away from the person stepping on it. Having traction prevents this rotation from happening. Key note: The center of mass of the person is actually separated at a distance from his/her foot on the banana. Since the line connecting that person's center of mass and the contact point is not usually aligned with the friction force between the ground and the banana, the friction on the ground can actually help counteract the torque on that person due to gravity and that contact point.

The centrifugal force (due to the tendency of traveling in a straight line) exists across the entire wheel, but the centripetal force imposed by the ground only exists as a counter-force to the bottom side of the wheel. They therefore help each other in countering the torque due to gravity and the normal force. The bicycle is able to trace a corner because the centripetal force on the bike (synonymous with the friction force by the ground to the bike in this case) is stronger than the centrifugal force on the bike (reaction to changes in velocity).

Therefore, a body in a turn can be both recipient of centrifugal and centripetal forces simultaneously. A common analogy you can witness on the road is called a rollover. The road provides a centripetal force due to friction, but the speed of the car overcomes this and the centrifugal force causes the vehicle to swerve outward, leading to a rollover with a direction of "rolling over" that depends on the torque caused by these forces.

Another thing that I am confused about is that you are talking about two forces acting on the bike: A centripetal force and a horizontal friction force. But these are the same force. The friction is what gives the bike a centripetal force. As far as I can tell, there are only three forces on the bike: The weight, the normal force and the friction.

The net effect of centripetal forces (friction in this case) and centrifugal forces (tendency to maintain a straight trajectory) on the wheel are what provide the necessary counterbalance needed to overcome the torque due to gravity and the "normal force" that would otherwise cause the object to tip over.

Thank you both for your explanations, I do appreciate your help very much.

You're very welcome!

 

[A.T.]

For simplicity you can consider a point mass on a mass-less beam attached via a hinge to a rotating beam.

In the rotating rest frame of the whole thing the torque from gravity around the hinge is constant, so if it wasn't canceled by the centrifugal torque, it would generate angular acceleration around that constant axis, and make the mass fall inwards.

In the inertial frame however the torque from gravity is a vector which continuously changes its direction, and that makes matters complicated. Simplistically: after half a circle the torque vector from gravity is flipped, so it cancels all the angular acceleration it created on the opposite side of the circle. But in reality this transition happens continuously.

This intuitive problem is similar to the question: Why doesn't the centripetal acceleration change the speed of an object in circular orbit, when it modifies the velocity vector? The reason is that the acceleration continuously changes direction to be perpendicular to velocity. But for torques it is ever more difficult to visualize.

 

[McLaren Rulez]

AT, that's a nice explanation. It took me a while but I can actually see what's going on now. Thank you very much!

 

[kmarinas86]

In the inertial frame however the torque from gravity is a vector which continuously changes its direction, and that makes matters complicated. Simplistically: after half a circle the torque vector from gravity is flipped, so it cancels all the angular acceleration it created on the opposite side of the circle. But in reality this transition happens continuously.

What has always bugged me with explanations like this is that they don't explain what a "rotating torque" vector is mechanically. A rotating momenta of inertia (mr^2) may lead to an angular momentum vector (mr^2/t), and a rotating angular momentum vector (mr^2/t) may lead to a torque vector (mr^2/t^2), but then what does a "rotating torque vector" lead to?

For me, these sorts of explanations serve as mystifying simplicity. It's like saying that "God did it". I understand that it is easier to "explain" things that way, but really, I don't see how it actually explains anything.

This intuitive problem is similar to the question: Why doesn't the centripetal acceleration change the speed of an object in circular orbit, when it modifies the velocity vector? The reason is that the acceleration continuously changes direction to be perpendicular to velocity. But for torques it is ever more difficult to visualize.

The problem with the "torque" of the precession is that it does not even lead to an angular acceleration. An angular acceleration implies a change in the magnitude of angular momentum. A precessing spin axis has nothing of the sort caused by the gravitational torque. The gravitational torque gives us the direction of the torque vector, but it does not give us the derivative of that torque vector, and thus by itself, it cannot say anything about how fast that vector must change direction.

From what I understand, the basic explanation could actually be simpler than what you gave. All one has to do is look at what a car does when it is moving too fast on a turn.



The spin of the top increases the tendency the travel in a straight line, and so it magnifies the ability to an continue outward path, via reactive centrifugal forces. This stabilizes the top because position and lateral acceleration of a spinning top are about 90 degrees out of phase as gravity tries to turn the top inwards.

 

[K^2]

McLaren, if you go to inertial coordinate system, the pivot point and CoM actually accelerate. If you pick a static pivot, you'll notice that you do get torque, but the angular velocity of CoM also changes.

The simplest solution for this is to use CoM as your pivot. That's what you normally do when you deal with free-bodies. In that case, acceleration of CoM is independent of angular motion of the body. Then looking at net torque produced by normal and centripetal forces, it is zero. And gravity can't produce torque because it acts on CoM. So that solves that.

 

[McLaren Rulez]

K^2, you're absolutely right. It is loads easier to consider the CoM or indeed switch to the frame of the bike itself. I was just curious why it seemed to be so hard in an inertial frame with the axis running through the contact point.

I see now that I failed to account for the fact that the circular motion of the bike gives us a change in angular momentum about my axis. I missed that term. And yet, the axis moves such that there is always a change in angular momentum about it but there is never any angular momentum. As A.T. said, that is analogous how there is a centripetal acceleration but never any centripetal velocity in uniform circular motion. That's quite cool now that I see it!

Thank you everyone for your replies; it's such a basic mechanics problem but I feel like I learned something new!