Alternative approach -- Bicycle on a curve

[Asad Raza]

Homework Statement

A bicycle of mass m is traveling at constant speed v around a curve of radius r without slipping. You can take the acceleration due to gravity as g. Calculate the angle of tilt, θ, that will enable it to balance.

Homework Equations

R=mg
Rsintheeta (Length)=Fcostheeta (Length); R is the normal force, F is the centripetal force and Length is the distance from the ground to the center pf mass
F=mv^2/r

The Attempt at a Solution

I solved this question by taking moments about the center of mass and using the above equations, and a bit of substitution. However, I have two queries:
i) Friction provides centripetal force and acts towards center (Sliding Friction. Right?). But friction should also act opposite to the direction of motion (tangent along the entire path). Why don't we account for it in the such problems. It is called rolling friction, I guess. Correct me if I am wrong.
ii) Secondly, if I take moments about the point of contact, instead of at the center of mass, the equation doesn't seem to balance as there's moment due to weight on one side of the equation and all the other moments turn out to be zero since their line of action passes through the pivot.

 

[haruspex]

Sliding Friction. Right?)

No. The tyres are in rolling contact, so static friction.

But friction should also act opposite to the direction of motion

Kinetic friction acts to oppose actual relative motion of surfaces in contact.
Static friction acts to oppose that relative acceleration that would occur were it not present.
Without friction, the bicycle would accelerate out radially relative to its actual path.

if I take moments about the point of contact, instead of at the center of mass, the equation doesn't seem to balance

The forward motion of the mass centre imbues it with angular momentum about the instantaneous point of contact. As a vector, that points radially. The torque generated by mg results in precession. In terms of the linear motion, that means the bicycle goes around the circle.

 

[CWatters]

But friction should also act opposite to the direction of motion (tangent along the entire path). Why don't we account...

The tyre isn't sliding or trying to slide in that direction so there is no friction force in that direction. There might be air resistance or rolling resistance but that's not mentioned in the problem so can be ignored.

 

[Chestermiller]

When you take moments about an axis that is not through the center of mass in an accelerating system, you need to include the moment of the pseudo force through the center of mass.

 

[haruspex]

When you take moments about an axis that is not through the center of mass in an accelerating system, you need to include the moment of the pseudo force through the center of mass.

Asad Raza took moments about the "point of contact". That is ambiguous because it is not clear whether it is the point on the road or a (moving) point on the tyre, but I took it to be the former. That makes it an inertial frame, so no pseudo forces.
Did you read my explanation in post #2?

 

[Chestermiller]

Asad Raza took moments about the "point of contact". That is ambiguous because it is not clear whether it is the point on the road or a (moving) point on the tyre, but I took it to be the former. That makes it an inertial frame, so no pseudo forces.
Did you read my explanation in post #2?

The above figure shows the forces acting on the bike with- and without the pseudo force shown. To do the moment balance correctly in an approach where the pseudo force is not included (in a problem involving accelerations), one must take moments about the center of mass. To do the moment balance correctly in an approach where the pseudo force is included, one can take moments about any point.

In either case, if moments are taken about the center of mass, the moment balance is given by:
$$RL\cos{\theta}=mgLcos{\theta}=\frac{mv^2}{R}L\sin{\theta}$$
This is the moment balance obtained by Asad Raza.

But when Asad Raza took moments about point A in the diagram to the left (without the pseudo force included), he obtained:

$$mgL\cos{\theta}=0$$This is, of course, incorrect.

But, if we take moments about point A in the diagram on the right (with the pseudo force included), we obtain:
$$mgL\cos{\theta}=\frac{mv^2}{R}L\sin{\theta}$$
This is in correct, and in agreement with the moment balance about the center of mass.

The key learning here is that, in a system that is not in equilibrium (i.e., involving acceleration), if you include the pseudo force, it doesn't matter which point you choose to take moments about. Any point other than the CM or point A would deliver the same result. But, if the pseudo force is not included, it is only valid to take moments about the CM.

 

[haruspex]

View attachment 207938
The above figure shows the forces acting on the bike with- and without the pseudo force shown. To do the moment balance correctly in an approach where the pseudo force is not included (in a problem involving accelerations), one must take moments about the center of mass. To do the moment balance correctly in an approach where the pseudo force is included, one can take moments about any point.

In either case, if moments are taken about the center of mass, the moment balance is given by:
$$RL\cos{\theta}=mgLcos{\theta}=\frac{mv^2}{R}L\sin{\theta}$$
This is the moment balance obtained by Asad Raza.

But when Asad Raza took moments about point A in the diagram to the left (without the pseudo force included), he obtained:

$$mgL\cos{\theta}=0$$This is, of course, incorrect.

But, if we take moments about point A in the diagram on the right (with the pseudo force included), we obtain:
$$mgL\cos{\theta}=\frac{mv^2}{R}L\sin{\theta}$$
This is in correct, and in agreement with the moment balance about the center of mass.

The key learning here is that, in a system that is not in equilibrium (i.e., involving acceleration), if you include the pseudo force, it doesn't matter which point you choose to take moments about. Any point other than the CM or point A would deliver the same result. But, if the pseudo force is not included, it is only valid to take moments about the CM.

Seems Chet and I disagree. Private conversation initiated.

 

[haruspex]

Seems Chet and I disagree. Private conversation initiated.

As a result of that private thread, I have developed all the equations for the point-of-contact reference frame, so I might as well post them here.

With the point of contact as origin, the centre of mass is at some vector $\vec c$. That means it has angular momentum $m\vec c\times\vec v$. The weight exerts torque $m\vec c\times\vec g$. This leads to a precession about a vertical axis.
If the precession rate is $\vec\omega$, the torque is $\vec \tau$ and the angular momentum is $\vec L$ then $\vec\tau=\vec\omega\times\vec L$.
Thus $\vec c\times\vec g=\vec \omega\times(\vec c\times\vec v) = \vec c(\vec\omega.\vec v)-\vec v(\vec\omega.\vec c)$.

Since $\vec\omega$ and $\vec v$ are orthogonal:
$\vec c\times\vec g=-\vec v(\vec\omega.\vec c)$
This is reasonable since $\vec v$ is indeed orthogonal to both $\vec g$ and $\vec c$.

The situation is analogous to force and linear momentum. To the extent that an applied force is parallel to the existing momentum it leads to a change in speed; to the extent that it is normal to the momentum it leads to a rate of change of direction (slew rate - I think that's the right word).

Similary, to the extent that a torque is parallel to the angular momentum it leads to a change in its magnitude; to the extent that it is normal it leads to a precession rate, i.e. the rate of change of direction of the angular momentum.
In the present case, the torque is normal to the angular momentum.​

Using c for $|\vec c|$ etc., theta for the tilt of the bicycle, R for the horizontal distance from the mass centre to the centre of the circle:
$\vec v =\vec \omega\times\vec R$
$v=\omega R$
$|\vec c\times \vec g|=cg\sin(\theta)$
$|\vec v(\vec\omega.\vec c)|=v\omega c\cos(\theta)$
Combining all those equations we find $v^2=Rg\tan(\theta)$

 

[Chestermiller]

Hi haruspex,

As I said in our private conversation, I was not able to follow what you have done. Any chance of providing a diagram.

One question, regarding $\theta$, that is the angle of tilt of the bike with respect to the horizontal in your analysis, correct?

 

[haruspex]

One question, regarding θ, that is the angle of tilt of the bike with respect to the horizontal in your analysis, correct?

No, to the vertical.

 

[Chestermiller]

No, to the vertical.

One more question. Is what I did incorrect? If so, why was I able to get the same answer as you?

 

[haruspex]

One more question. Is what I did incorrect? If so, why was I able to get the same answer as you?

You effectively used a non-inertial frame. To ignore the precession, your point of contact axis must be moving with the bicycle. Thus the pseudo-force was necessary.

 

[Chestermiller]

You effectively used a non-inertial frame. To ignore the precession, your point of contact axis must be moving with the bicycle. Thus the pseudo-force was necessary.

Is that an incorrect thing to do?

 

[haruspex]

Is that an incorrect thing to do?

Not at all, but as I wrote in post #5 it is not entirely clear whether @Asad Raza was taking the "point of contact" frame of reference as the instantaneous point on the ground (i.e. a fixed point, so an inertial frame), or as the moving point of contact. And, as I wrote there, I assumed Asad was taking the former view and could not understand how there could be an unbalanced torque. In that view, the explanation is that there is an unbalanced torque, but the consequence of it is precession, not a change in tilt of the bicycle.
If one takes the moving point of contact as axis then there is a pseudo force to be included, which is the way you approached it.

As far as right or wrong is concerned in the context of this thread, it depends on which view Asad intended. If Asad intended the inertial frame then your approach, while valid in itself, did not address the question posed.