Physics Quiz 3

[Ether]

Originally Posted by John

so the total friction force would be ... tau_k/(r*sin(theta))

Yes.

But according to the Coulomb friction model, the total friction force must be mu_k*W.

So that means tau_k must equal mu_k*W*r*sinθ.

So, what happens if the voltage to the motor is increased?

[BRAVESaj25bd8]

Originally Posted by Ether

Yes.

But according to the Coulomb friction model, the total friction force must be mu_k*W.

So that means tau_k must equal mu_k*W*r*sinθ.

So, what happens if the voltage to the motor is increased?

This doesn't sound right to me. The torque applied by the motor should not be changing as theta changes. I think the problem is that in order to find the friction equal to tau_k/(r*sin(theta)), you must assume the wheel is gripping. This does make sense because it is able to rotate and drive around the pivot but is inconsistent. I feel like one of the boundary conditions is actually overconstraining the model.

[Ether]

Originally Posted by BRAVESaj25bd8

This doesn't sound right to me. The torque applied by the motor should not be changing as theta changes

tau_k is the motor torque necessary to be in equilibrium at a given angle theta, given the assumptions in the model (mu_k constant, frictionless pivot, frictionless wheel bearing, no wind resistance, no rolling friction, no friction due to rotation of the wheel about the Z axis).

What the equation is saying is that if you reduce the angle theta, it takes less tau_k to be in equilibrium. If you increase theta, it takes more.

Sustaining equilibrium when theta is very small takes very little torque. In fact, when theta is zero it takes zero torque to sustain equilibrium since the pivot is frictionless and we are ignoring wind resistance, rolling friction, friction in the wheel bearings, and friction due to rotation of the wheel about the Z-axis.

So what happens if you start with the system in equilibrium, and you then increase tau_k to tau_k' ?

[BRAVESaj25bd8]

After thinking about it more, the error I made was getting wrapped up in angular motion, thinking that theta was the wheel's angular position about the pivot as the system rotates. What it really represents, though is a constant in this problem, the angle between the wheel's axis of rotation and the pivot arm.

I think that if you increase the torque, nothing will change because the wheel is already slipping. The friction cannot be higher than W*mu so no more torque is able to be useful.

[Ether]

Originally Posted by BRAVESaj25bd8

I think that if you increase the torque, nothing will change because the wheel is already slipping. The friction cannot be higher than W*mu so no more torque is able to be useful

If you increase tau_k, the system will no longer be in equilibrium. That means something must change.

[Ninja_Bait]

Originally Posted by Ether

If you increase tau_k, the system will no longer be in equilibrium. That means something must change.

Can normal force increase to balance the increase in tau_k? I don't think that actually makes sense but it seems to be the only variable that can change in this case.

[Ether]

Originally Posted by Ninja_Bait

Can normal force increase to balance the increase in tau_k?

No.

I don't think that actually makes sense but it seems to be the only variable that can change in this case.

If the system is in equilibrium, and then you increase the motor torque that's driving the wheel, what happens?

Think torque=momentOfInertia*angularAcceleration. Then follow the causes and effects as they ripple through the system.

[James Critchley]

Ninja_Bait: Ether has already answered this question in an earlier post... I think the issue is the notion of supplying a larger torque indefinitely. Perhaps it will help to say it another way?

The system will accelerate to a speed where you are no longer able to supply torque greater than tau_k. Physical systems are power limited and power = torque * angular rate. The system will accelerate to the limits of your power supply or motor torque vs speed curve (which ever comes first). At that point the torque supplied by the motor will again be tau_k and you will have constant speeds.

If you could somehow supply a constant torque independent of speed, the system as modeled would properly accelerate forever. Other effects would eventually dominate and the model would become invalid so this wouldn't actually happen, but you would still want to patent that motor and power supply!

[Ninja_Bait]

Ah, I see. Thank you.