Does a box of Pendulum's weigh more if they are swinging?

Question

The question is more specifically to explain why the box doesn't weigh more, because if it did then I could revolutionize space travel.

Consider a rigid box suspending two counter oscillating pendulums of mass m/2. Starting against the top, at the bottom of their swing the masses will be moving with a speed of $\sqrt{2gl}$ where $l$ is the length of the pendulum. The sum tension in the lines at this moment should be equal to the weight of the masses plus the centrifugal force: $T = mg + \frac{mv^2}{r}$. Plugging in the speed at the bottom gives a tension of $T = 3mg$. I figure the weight of the full box is the weight of the box material plus the downward component of the tension, as any horizontal components are canceled by the counter-oscillating setup.

So it seems at the bottom of the pendulums swing, the box weighs more than when with stationary pendulums. And, intuitively, the box will weigh less as the tension goes to zero at the apex.

I can envision a box suspending two sets (four total) of counter rotating powered rods that are out of phase such that at any moment there is a non-zero downward component of tension, making this box always heavier when the rods are moving.

Why couldn't I use this "extra weight" to transfer momentum?

Answer 1

The double pendulum apparatus has an apparent weight equal to the sum of the weights of the box and two pendulums when at rest, and weighs more than that at the bottom of the swing, but weighs less than that at the top of the swing. Also note that the bob moves through the bottom of the swing very quickly, meaning there is a very brief time when the apparent weight increases, but that the bob spends comparatively more time at the extremes of the swing when its velocity is low and apparent weight is less than its true weight.

As such, it is not the case that starting two double pendulum apparatuses will make a box appear heavier overall at all times. The key is that you need to time-average the effects of the pendulums - while one very briefly makes the box heavier, the other is making the box lighter, by a smaller amount but for a longer time. The apparatus' apparent weight oscillates between slightly heavier and slightly lighter at any instant of time as the pendulums swing, but on average, is no different than with the pendulums at rest.

It is possible to transfer momentum out of a system like this, but you're just decreasing the amount of energy in the system, and will eventually run it down. You could transfer some momentum out of the box at the bottom of a pendulum swing, but the pendulum will of course not continue to swing with the same amplitude. You don't get energy "for free" by starting a double pendulum and having the box magically increase in weight in the absence of other external forces.