Conservation of mechanical energy and external forces. A bit confused

[eventob]

Hi

I got a question regarding conservation of mechanical energy. The mechanical energy of a system is conserved, as long as there are no transfer of energy (like heat, work, mechanical waves etc.), right? So, why is the mechanical energy in the system of the pivoting rod and the earth, as shown in the picture below conserved? Isn't there are a force acting from the pivot point, doing work?

I wasn't sure whether I should post it here, or in the homework section, but I figured since it was a general problem relating to mechanics, it was better to post it here.


Thanks in advance

 

[K^2]

Work is force integrated over displacement. No displacement, no work.

However, if you go into moving coordinate system, where pivot is moving relative to observer, the pivot does do work. In that coordinate system, however, the total energy of the rod does change.

Momentum is more interesting, since momentum is force integrated over "displacement" in time. That means that momentum is transferred from rod to the pivot in any coordinate system.

 

[AndyNewton]

Start by considering what happens if the rod is not connected to the Earth at a pivot point.

The system of the Earth and the rod has gravitational potential energy when the two objects are separated in space and they have the er.. "potential" to fall towards each other.
When such a rod does fall freely downwards the Earth also falls upwards towards the rod. The system overall loses potential energy and gains an equal quantity of kinetic energy. Total energy is conserved. Virtually 100% of the kinetic energy goes into the rod.

-------------------------------------------------------------------------------------------------------------------------------

Now attach the rod to the pivot as shown in diagram message #1. The upright section supporting the pivot must be in contact with the earth, not shown in the diagram but assumed.

This is a more complex situation. The free end of the rod is similar to the entirely free unconnected rod as described above. But the pivot end of the rod is similar to an object resting on a fixed platform - where gravitiational potential energy remains constant because nothing moves. The overall system is therefore a mixture of those two conditions.

Conclusion:
When released, the rod and Earth system do experience a partial movement towards each other but the movement is somewhat limited by the support at the pivot end of the rod. Despite that limited movement however, there is nothing really strange happening. The rod is falling a small distance towards the Earth whilst gaining kinetic energy. Also the Earth "falls" an unmeasurably small distance upwards (lifting the pivot).

And... in the pivot case, the Earth must experience a sideways force while the rod is swinging downwards, transmitted as a tension force through the pivot. So angular momentum is also conserved. When the rod is swinging through its lowest position the Earth's rotation is changed by an unmeasurably small angular speed in the opposite direction. That's nice!

 

[K^2]

You don't need to make it that complex. You can assume that the support is infinitely massive and does not move in response to rod's movement. There is still nothing weird happening with the energy.

 

[PJC01]

for your help!

Hello,

Thank you for your question. The conservation of mechanical energy is a fundamental principle in physics that states that the total energy of a closed system remains constant over time. This means that the sum of the kinetic and potential energy of all the objects in the system remains constant, as long as there is no external transfer of energy.

In the situation you described, the pivoting rod and the earth can be considered as a closed system. The force acting from the pivot point does not affect the conservation of mechanical energy because it is an internal force within the system. Internal forces do not transfer energy in or out of the system, so they do not affect the total mechanical energy of the system.

The work done by the force at the pivot point may change the kinetic and potential energy of the individual objects in the system, but the total mechanical energy remains constant. This is because the work done by the force is equal and opposite to the change in kinetic and potential energy, as described by the work-energy theorem.

I hope this helps to clarify the concept of conservation of mechanical energy and how it applies to external forces. Please let me know if you have any further questions.