Imagine a pendulum swinging back and forth. When it is at its lowest point the PE = 0 and KE is maximal. Now imagine an observer moving along in the direction of the pendulum, with matched velocity at the lowest point of the pendulum. From the observer's viewpoint the PE is apparently zero, but the velocity and hence the KE is also zero.
From whence did the energy go? What is the correct way to do an energy balance for this case?
A system of pendulum consists in truth of two bodies, the pendulum itself and the planet earth. Otherwise there is no momentum conservation because the pendulum moves back and forth. While the pendulum moves forth, the earth moves back with a very small velocity so that momentum is conserved. If you move along with the forth velocity of the pendulum, you experience a greater back velocity of the earth.
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