The Dubai resort at which you recently accepted a job is losing business to a co

Question

The Dubai resort at which you recently accepted a job is losing business to a competitor with a tankball facility, and your boss has decided that your resort needs an unprecedented new thrill ride. You meet with your boss and with your new assistant to discuss the idea. Your boss has produced the following sketch: The plan, he explains, is to build an enormous pendulum with a sliding carriage. Thrill-seekers will be strapped into the carriage near the top of the pendulum arm with the arm raised - in other words, with z = 1 meter and theta = pi /2 radians initially - and then released from rest. As the arm swings down under the weight of the carriage, the carriage will slide outward along the arm. When the carriage reaches the end of the arm, it will be flung through the air into the Persian Gulf. It's clear, your boss explains, that the carriage can be regarded as a point mass and that the pendulum arm can be regarded as massless in modeling this system. The system will be governed by a pair of differential equations coupling the evolution of z(t) with the evolution of theta (t). At this point, your assistant pipes up. "One of the governing equations will describe the evolution of angular momentum about the pivot point," he squeaks, "while the other will describe the dynamics of the carriage sliding outward along the arm." Your boss looks pleased, so your assistant continues. "I can even figure out the second equation by inspection," he announces. "Clearly, m = mg cos theta according to Newton's second law!" Your boss frowns. "You're right about the first equation, but the second equation of motion should be m - mz = mg cos theta." What was wrong with your assistant's analysis of the motion of the carriage along the arm? The net force on the carriage in this direction is clearly mg cos theta. Assuming your boss is correct, what are the two ODEs governing the motion of this system? Roughly how long will the arm need to be to guarantee that the carriage will remain on the arm for three seconds? Does this depend on the mass of the carriage? Is it possible (assuming that the carriage slides down the arm frictionlessly) to choose the mass of the carriage and the length of the arm so that theta will swing negative before the carriage is released?

Explanation / Answer

1) assistant did not account for change in theta with time.

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