Pendulum rocket fallacy

Robert Goddard next to the first liquid-fueled rocket, 1926

The pendulum rocket fallacy is a common fundamental misunderstanding of the mechanics of rocket flight and how rockets remain on a stable trajectory. The first liquid-fuel rocket, constructed by Robert Goddard in 1926, differed significantly from modern rockets in that the rocket engine was at the top and the fuel tank at the bottom of the rocket. It was believed that, in flight, the rocket would "hang" from the engine like a pendulum from a pivot, and the weight of the fuel tank would be all that was needed to keep the rocket flying straight up. This belief is incorrect—such a rocket will turn and crash into the ground soon after launch, and this is what happened to Goddard's rocket. Use of basic Newtonian mechanics shows that Goddard's rocket is just as unstable as when the engine is mounted below the fuel tank, as in most modern rockets.[1]

Practical explanation

No rocket can be constructed perfectly. Inevitably, the engine's direction of thrust will not be perfectly aligned with the rocket's center of mass, so the rocket will have an inbuilt tendency to turn. When this happens, the engine starts rotating with the rest of the rocket (regardless of its shape) and the direction of thrust rotates as well. Except for air resistance, there is no rotational force or torque available to turn a simple aerodynamic rocket back onto its correct path, as can be shown from the classical Newtonian physics reasoning in the next section. Consequently, the initial deviation from a vertical path will increase over time, and a rocket constructed in this way will always turn around and strike the ground eventually.

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,[2] 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.

Solutions

Test of the Launch Escape System used on the Apollo spacecraft

To fly correctly, rockets must have a means of stability. The fins of model rockets and the sticks of firework rockets act aerodynamically to keep the axis of the rocket pointing in the direction of flight, which will not overcome the instability, but will greatly reduce it, and will give a gravity turn trajectory. Larger rockets can do without fins by using a guidance and control system that actively steers the rocket and keeps it flying in the intended direction.

Even a Goddard-type rocket, with the engine at the front, will fly correctly if fitted with fins or another means of control. Examples of this are the launch escape systems fitted to some crewed spacecraft such as the Saturn V rocket.

References

1. ^ "Jim Bowery". "The Pendulum Rocket Fallacy". Archived from the original on 2009-10-18. Retrieved 2010-06-30.
2. ^ See explanation in the center of mass article.