Pendulum Rocket Fallacy - Physical Reasoning

Physical Reasoning

The pendulum belief is a fallacy because it stems from the implicit (and incorrect) assumption that simply because the weights and 'hanging' devices are arranged in roughly the same way in both a rocket and a pendulum, they will behave in the same fashion. The forces exerted are, however, different. While gravity does act similarly in both physical systems, the supporting force exerted onto the pendulum by its hanging point is constrained to remaining aligned with the fixed point; this is unlike the force exerted onto the rocket by its engine, whose direction instead depends on the rocket's overall orientation or attitude.

The physical system constituted by a rocket, like Goddard's, comprises the engine, tank, and rigid frame. Assuming that air resistance is not significant, there are only two forces exerted on the system as a whole: (1) gravity, and (2) the reaction force caused by the ignited gases being expelled from the rocket's nozzle at high speed. Examining the moment of each of these forces with respect to the center of mass of the system:

Gravity

The center of gravity is identical to the center of mass and therefore gravity does not exert any torque. This is a general property of all systems in a uniform gravitational field.

Reaction force from the engine

Due to the rigid construction of the rocket frame, the force is exerted on a line that is fixed with respect to the rocket. The unavoidable imperfection mentioned above means that this line does not contain the center of mass precisely. The amplitude of the reaction force depends on the thrust of the engine, which is always positive. The torque is, therefore, exerted with respect to an axis whose direction is fixed with respect to the rocket frame, and is of constant sign.

Given that torques are pseudo-vectors, and hence add linearly, it follows that the rotation speed of the rocket around the aforementioned axis can only increase.