Let's say we have a pendulum and with an initial displacement of 5 degrees, it starts oscillating. Assuming ideal conditions, the pendulum will oscillate forever with its maximum sway angle being 5 degrees. Is it possible possible to bring the pendulum to a stop or reduce the maximum sway angle by just changing the length of the pendulum while in motion?
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2 Answers
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The first question is: Does your pendulum have a string or a rod? If you were to, say, suddenly introduce 20 meters of string without moving the mass at the end, it will fall straight down, and then the behaviour can be all sorts of things depending on how elastic the string is.
If it is a rod (and so the mass moves with the added length), then it depends on when you do the lengthening. If you do it at a moment with maximum potential energy (and no kinetic energy), the amplitude stays the same. If you do it at a point of minimum potential energy (the bottom, with maximum kinetic energy), then the amplitude will decrease, but never to zero (since the bob is still moving). You can get amplitude arbitrarily low, but not to zero.
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For a pendulum with a massless rod (see here):
Period of oscillation (small angle $\theta$):
$$T\approx 2\pi\sqrt{\frac{L}{g}}$$
Hamiltonian (total energy):
$$H=K+U= \frac{1}{2}mL^2\dot\theta^2 + mgL(1-\cos\theta)$$
So $H$ depends both on $\theta$ and $L$.
Now if we had a way of increasing $L$ without using some external force (so that no energy is added or subtracted from the system), then $H$ must remain the same. This means that if we increase $L$ in this way, $\theta$ must be adjusted too (to a lower value)
