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A ring of mass M rests on a smooth horizontal surface and is pinned at a point on its circumference so that it is free to swing about a vertical axis. A bug of mass m crawls around the ring with constant speed. a) Set up the equations of motion, taking this as a system with two degrees of freedom, with the force exerted by the bug against the ring to be determined from the condition that he moves with constant speed. b) Now set up the equation of motion, taking this as a system with one degree of freedom, the bug being constrained to be at a certain point on the ring at each instant of time. Show that the two formulations of the problem are equivalent.

A ring of mass M rests on a smooth horizontal surface and is pinned at a point on its circumference so that it is free to swing about a vertical axis. A bug of mass m crawls around the ring with constant speed. a) Set up the equations of motion, taking this as a system with two degrees of freedom, with the force exerted by the bug against the ring to be determined from the condition that he moves with constant speed. b) Now set up the equation of motion, taking this as a system with one degree of freedom, the bug being constrained to be at a certain point on the ring at each instant of time. Show that the two formulations of the problem are equivalent.

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Derive using the Lagrangian and Lagrange's equation only.

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A ring of mass M rests on a smooth horizontal surface and is pinned at a point on its circumference so that it is free to swing about a vertical axis. A bug of mass m crawls around the ring with constant speed. a) Set up the equations of motion, taking this as a system with two degrees of freedom, with the force exerted by the bug against the ring to be determined from the condition that he moves with constant speed. b) Now set up the equation of motion, taking this as a system with one degree of freedom, the bug being constrained to be at a certain point on the ring at each instant of time. Show that the two formulations of the problem are equivalent.
Transcribed Image Text:A ring of mass M rests on a smooth horizontal surface and is pinned at a point on its circumference so that it is free to swing about a vertical axis. A bug of mass m crawls around the ring with constant speed. a) Set up the equations of motion, taking this as a system with two degrees of freedom, with the force exerted by the bug against the ring to be determined from the condition that he moves with constant speed. b) Now set up the equation of motion, taking this as a system with one degree of freedom, the bug being constrained to be at a certain point on the ring at each instant of time. Show that the two formulations of the problem are equivalent.
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