Stopping a rod moving with constant omega

[SampleLow]

Homework Statement

A rod of mass m and length l is hinged at one of its ends. The rod is then rotated with angular velocity ω which remains constant because it is powered. The ground that the rod is placed on has friction coefficient μ. Find at a distance x, from the hinge, the minimum force F that must be applied at that point vertically so that the rod stops moving. (The force is applied perpendicular to the ground and is parallel to mg.)

Relevant Equations

f=μmg.
dM=N1-N2
N1+N2=mg+F.

Friction before force is applied, f=μmg.
After force is applied on element dx, at a distance x from hinge, the is a bending moment on that element dM which is given by normals on either side (say N1, N2) by dM=N1-N2 and N1+N2=mg+F.

 

[PeroK]

I think you have misinterpreted the question. The force $F$ is to be applied at one particular point a distance $x$ from the hinge.

 

[SampleLow]

I think you have misinterpreted the question. The force $F$ is to be applied at one particular point a distance $x$ from the hinge.

What do you mean? If it is applied at one point, then that element will have a bending moment. What is incorrect?

 

[PeroK]

What do you mean? If it is applied at one point, then that element will have a bending moment. What is incorrect?

The problem has nothing to do with mass elements or bending moments.

 

[SampleLow]

The problem has nothing to do with mass elements or bending moments.

Then what is it about?

 

[PeroK]

Then what is it about?

Friction.

 

[SampleLow]

Ok. So when I add a force F, the friction becomes μ(mg+F).

 

[PeroK]

Ok. So when I add a force F, the friction becomes μ(mg+F).

Sort of. The force of gravity is spread across the length of the rod. The force $F$ is localised at one point.

Plus, gravity is irrelevant. Can you see why?

 

[SampleLow]

I don't really understand.

 

[PeroK]

I don't really understand.

I can't give you the answer. Suppose there was a real, large, long rod being forcibly rotated and you had to try to stop it (or at least slow it down) by sitting on it and pushing it into the ground with your weight. Where would you sit on it and why?

 

[PeroK]

Is that the whole problem statement? I just realised that if the torques on the rod are balanced then any additional torque will stop it eventually. The problem says nothing about how quickly the rod must stop.

 

[haruspex]

There is so much wrong with the question.

First, what are we to suppose are the characteristics of the driving torque? Is it constant torque, constant power, or something else?
Since we are given $\omega$, we can derive an expression for the initial power, but if the power is constant then increasing the frictional torque will only slow it, not stop it.
On the other hand, as @PeroK notes, if constant torque then any nonzero $F$ will do it.

Secondly, how are we to suppose the normal force from the ground is distributed when $F$ is applied? Taking the resulting extra normal force $-F$ to be entirely at distance $x$ from the axle is rather unrealistic. It would require the rod to be like a section of bicycle chain, bending freely in the vertical plane but rigid in the horizontal plane.

 

[SampleLow]

It is possible the question is flawed.

Where would you sit on it and why?

The other end.

any additional torque will stop it eventually

Would the normal reactions on either side not change suddenly?

if the power is constant then increasing the frictional torque will only slow it, not stop it.

I was assuming constant power. But why will it only slow down?

Secondly, how are we to suppose the normal force from the ground is distributed when $F$ is applied? Taking the resulting extra normal force $-F$ to be entirely at distance $x$ from the axle is rather unrealistic.

My doubt exactly.

It would require the rod to be like a section of bicycle chain, bending freely in the vertical plane but rigid in the horizontal plane.

What is this? Could you give me an example of such a problem? Maybe just link it.

 

[haruspex]

I was assuming constant power. But why will it only slow down?

Just like $P=Fv$, $P=\tau \omega$. Increasing $\tau$ reduces $\omega$ but never drives it to zero.
In the real world, motors have both a maximum power and a maximum torque.

Could you give me an example of such a problem?

The example I gave, a section of chain from an engine; bicycle chain, motorcycle chain, …
Anything rod-shaped that is fairly inflexible in one plane but highly flexible in another.