Misunderstanding Work-energy theorem and center of mass properties

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. 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????

• Voting to reopen. This is a perfectly valid conceptual question about why constraint forces do no work. Nov 2, 2023 at 12:58

Remember that the hinge is fixed and assumed to be frictioneless.

Free body diagram for the rod.

Ask yourself, "Does the force $$X$$ move along its line of action?"
If the answer is "No.", then the force $$X$$ does no work.

So the situation that we have here, is a rod that is parallel so there is a force down due to gravity, and then there is a force that becomes important as the rod starts rotating because initially the force is essentially useless when the rod is parallel. The force is essentially diagonal, so essentially we have two important forces here. One due to gravity and one due to the hinge.

The reason is because as the object as described rotates, the hinge force while external is also perpendicular to the direction of motion(velocity), and assuming the force from the hinge is constant, it means that the rod will only change direction, and the hinge force changes direction accordingly as it the rod moves so the force is always perpendicular.

So W=Fdcos(theta), means that since theta is 0, it means Work done is zero. So it means all the work is done by gravity, which makes sense because there is potential energy stored in the gravitational field which can be used for the block's kinetic energy but the hinge itself doesn't seem to have an energy source(unless if it had some kind of motor-in which case the problem might be a bit different).

So essentially the concept is that perpendicular forces produce no work, no matter if it is external or internal.

• I get that there is a component of hinge force along the rod but you ignored component of recation force perpendicular to the rod and that force does work because costheta is -1. So there is a work done by reaction force. Nov 2, 2023 at 13:52
• 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 negative. Nov 2, 2023 at 13:58