Misunderstanding Work-energy theorem and center of mass properties

[physicsissohard]

Homework Statement

A rod is hinged at one of its ends, and released from rest when it is held parallel to the ground. The question is to find the angular velocity after the rod makes 60 degrees with the horizontal.

Relevant Equations

MgL/2=Ialpha a=r*alpha

To do this apparently, you need to use the work-energy theorem. You can calculate work done by gravity easily. However it was said that work done by the reaction forces from the hinge is zero, I don't get why.

Reaction Force from the hinge is an external force on the rod, and all external forces act on the center of mass and contribute to the displacement of the center of mass, so how is work done by reaction forces zero?

For example, take the initial position at t=0. Then If we consider torque about the hinge we get angular acceleration as MgL/2=Ialpha and alphar is the acceleration of the center of mass. and using newtons second law, Mg-R=Ma we found a and reaction force turns out to be Mg/4. So there is a reaction force perpendicular to the rod, at all times so work is not perpendicular so work done is not zero

 

[berkeman]

However it was said that work done by the reaction forces from the hinge is zero, I don't get why.

Work is force exerted through a distance. If the friction is zero at the hinge, what is the distance that a force is exerted through?

 

[PeroK]

Homework Statement: A rod is hinged at one of its ends, and released from rest when it is held parallel to the ground. The question is to find the angular velocity after the rod makes 60 degrees with the horizontal.
Relevant Equations: MgL/2=Ialpha a=r*alpha

To do this apparently, you need to use the work-energy theorem. You can calculate work done by gravity easily. However it was said that work done by the reaction forces from the hinge is zero, I don't get why.

Reaction Force from the hinge is an external force on the rod, and all external forces act on the center of mass and contribute to the displacement of the center of mass, so how is work done by reaction forces zero?

For example, take the initial position at t=0. Then If we consider torque about the hinge we get angular acceleration as MgL/2=Ialpha and alphar is the acceleration of the center of mass. and using newtons second law, Mg-R=Ma we found a and reaction force turns out to be Mg/4. So there is a reaction force perpendicular to the rod, at all times so work is not perpendicular so work done is not zero

If you don't believe the work energy theorem, then calculate the angular velocity using torque about the hinge. Then check against the work energy theorem and see how much work is done by the force at the hinge. The change in gravitational potential energy should be straightforward to calculate.

 

[physicsissohard]

If you don't believe the work energy theorem, then calculate the angular velocity using torque about the hinge. Then check against the work energy theorem and see how much work is done by the force at the hinge. The change in gravitational potential energy should be straightforward to calculate.

Seriously, What do you mean I don't believe work energy theorem? I just doubt if the application of WET as shown is correct or not? and there is no way you can calculate angular velocity using torque alone, well you can but that requires you to solve an unsolvable differential equation with elliptic integral and stuff like that.

 

[Doc Al]

For example, take the initial position at t=0. Then If we consider torque about the hinge we get angular acceleration as MgL/2=Ialpha and alphar is the acceleration of the center of mass. and using newtons second law, Mg-R=Ma we found a and reaction force turns out to be Mg/4. So there is a reaction force perpendicular to the rod, at all times so work is not perpendicular so work done is not zero

Sure, there is a non-zero reaction force at the hinge. As pointed out, that force does no real work since it has no displacement.

You could use Newton's 2nd law to calculate the velocity of the center of mass as the rod falls to the 60-degree position. But that will involve some tricky integration. (Note that the reaction force is not constant as the rod falls.)

Much easier to use conservation of energy. The only force acting on the rod that does work is gravity, so you only need to consider changes in gravitational energy.

 

[Steve4Physics]

To do this apparently, you need to use the work-energy theorem.

Using the work-energy theorem is not essential - but it is the simplest approach here.

You can calculate work done by gravity easily. However it was said that work done by the reaction forces from the hinge is zero, I don't get why.

The reaction force (from the hinge on the rod) acts on (say) point A on one end of the rod. Point A has zero displacement - so what can you say about the work done by the reaction force?

Reaction Force from the hinge is an external force on the rod, and all external forces act on the center of mass

Not sure what 'act on the center of mass' means.

A force does not have to do work on an object to affect the motion of the object. E.g. consider a mass moving in a circle at contant speed; the centripetal force (e.g. tension in a rope) causes acceleration but does no work.

Here the reaction force affects the rod's motion - but that doesn't mean the reaction force does work.

For example, take the initial position at t=0. Then If we consider torque about the hinge we get angular acceleration as MgL/2=Ialpha and alphar is the acceleration of the center of mass. and using newtons second law, Mg-R=Ma we found a and reaction force turns out to be Mg/4.

Yes. The reaction force exists (and is initially Mg/4). But it doesn't do any work (see above).

So there is a reaction force perpendicular to the rod, at all times

No. For example if the rod eventually reaches it's lowest point, the reaction force acts upwards (parallel to rod).

so work is not perpendicular so work done is not zero

Don't know what that means!

[Minor edits made]

Not, if the reaction force did work here, it would have to have a source of energy - but it doesn't!

 

[PeroK]

⁹Seriously, What do you mean I don't believe work energy theorem? I just doubt if the application of WET as shown is correct or not? and there is no way you can calculate angular velocity using torque alone, well you can but that requires you to solve an unsolvable differential equation with elliptic integral and stuff like that.

You're right. I should have said if you don't believe the force at the hinge does no work.

You don't have to solve an elliptic integral. You could look at the rate of change of KE using torque and rate of change of GPE and check the two are equal and opposite:
$$\tau = \frac{mgl}{2}\cos \theta$$$$\ddot \theta = \frac{\tau}{I} = \frac{mgl}{2I}\cos \theta$$
$$KE = \frac 1 2 I\dot \theta^2$$$$\frac {dKE}{dt} = I\dot \theta \ddot \theta = \frac{mgl\dot \theta}{2}\cos \theta$$
$$PE = -\frac{mgl}{2}\sin \theta$$$$\frac{dPE}{dt} = -\frac{mgl}{2}(\cos \theta)\dot \theta$$Hence the change in KE caused by the torque is entirely accounted for by the change in gravitational PE. The force at the hinge, therefore, does no work.

 

[kuruman]

Reaction Force from the hinge is an external force on the rod, and all external forces act on the center of mass and contribute to the displacement of the center of mass, $\dots$

That's a serious misconception that needs to be remedied. Yes, the force from the hinge is an external force. Yes, it is part of the net force acting on the rod and contributes to the acceleration of the center of mass. No, it does not act on the center of mass. It acts at the tip of the rod where it is attached to the hinge.