Pendulum: Energy is conserved but not momentum

[greypilgrim]

Hi.

In an ideal pendulum, energy is conserved. Potential energy gets transformed to kinetic energy and vice versa. However, momentum is not conserved.

The latter means that the pendulum is not an isolated system, which is plausible, since gravity is an external force. But why is energy conserved then? Under what conditions does an external force not change the energy in a system?

 

[S.G. Janssens]

Under what conditions does an external force not change the energy in a system?

When the force is conservative, i.e. dependent on position only. (This may sound like a tautology, but it is not.) Gravity is a conservative force.

 

[vanhees71]

In addition the force also must have a potential to be conservative in the usual sense. The Langrangian/Hamiltonian is not explicitly time-dependent and thus energy is conserved (Noether's theorem applied to time-translation invariance).

 

[PeroK]

I would say that energy is not conserved. The Earth not only gains and loses some momentum but also some kinetic energy. The momentum is significant but the energy is not. Effectively, therefore, energy of the pendulum appears to be conserved.

Consider also an elastic collision between a ball and the ground. Momentum of the ball is clearly not conserved but kinetic energy appears to be. Can you explain why that is?

 

[geir arge norway]

We can say that the potentian energy of the pendulum is stored in the gravitational field.
When the pendulum is moving upwards, energy is leaving the pendulum and stored in the gravitational field. This is due to the work done by the gravitational force. When the pendulum is moving downwards, energy is leaving the gravitational field and put into the pendulum. Again due to the work done by the gravitational force.
In this sense we can say that the external force changes the energy of the pendulum all the time.

 

[geir arge norway]

We can add that when energy is stored in the gravitational field, it is still belongs to the the pendulum. It is like having money in the bank and not in your wallet. Your money in the bank is still your money.