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I'm trying to wrap my head around a conceptual problem involving a simple pendulum with a rocket attached to its mass. Imagine the rocket expels gas to provide a tangential thrust force. However, the thrust is calibrated such that it's not sufficient to swing the pendulum upward; instead, it only counteracts the gravitational pull, bringing the pendulum to a point of equilibrium (let's say, 45°) where the downward gravitational force equals the upward thrust.

Under ideal conditions, where we negate the effects of friction, air resistance, and other non-conservative forces, I can't seem to understand the conservation of energy in this system. The rocket's fuel conversion into kinetic energy of the expelled gases generates thrust that maintains this equilibrium. Yet, with the pendulum statically held in this equilibrium state:

Where does the energy imparted by the thrust go, considering the pendulum itself does not seem to gain kinetic or potential energy?

Qmechanic
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Eduardo González
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3 Answers3

23

If the pendulum is in equilibrium then the rocket motor does no work on the pendulum. It exerts a force on the pendulum, but because the pendulum is not moving, this force does no work on the pendulum. It is exactly as if the pendulum was held by a length of rope - the rope exerts force on the pendulum but does no work on it.

The rocket motor, of course, does work by expelling its exhaust, but the energy that goes into the exhaust is initially seen as kinetic energy of the exhaust, and is eventually dissipated into the environment as sound and heat.

Note that during the initial phase of the motion - as the rocket motor moves the pendulum from vertical to its new equilibrium position - the velocity of the exhaust is lower than in the equilibrium position. This is because the exhaust has a fixed velocity relative to the rocket motor, which is now moving. So in this initial phase the rocket motor does less work on the exhaust and does some work on the pendulum instead - and this energy is stored as the potential energy of the pendulum in its new equilibrium position.

gandalf61
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All of the energy of the rocket motor in your case goes into moving air/used propellant downward. None of it goes into adding velocity to the rocket or pendulum. Eventually the velocity of that air/propellant starts to get diffused as eddies form or the plume interacts with the ground, and that kinetic energy in the air dissipates, converting into heat energy.

The situation you have could be simplified. The image of a rocket on a pendulum is rather gratifying, but if it is at equilibrium with the forces of gravity, the force diagrams are comparable to a static fire of a rocket engine. In that case, instead of working in a delicate equilibrium, we simply clamp the rocket down and light it. I find the visuals of a static fire do a good job of demonstrating where the energy goes.

Cort Ammon
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Hovering

Your scenario is equivalent to a rocket suspended above a suitable gravitational body, firing its engines so that it may remain in place. Of course, the gravitational field is accelerating the rocket towards the surface, so the engine must exactly counteract that acceleration. Thus, the downward momentum of the exhaust must perfectly balance the momentum of the rocket in free-fall. The COM of the rocket + exhaust system is in free-fall, conserving the energy expended. The pendulum here, is a mere distraction.

Lawnmower Man
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