Questions: A vibrating system made of two pulley fixed rigidly to each other to

Question

Questions:

A vibrating system made of two pulley fixed rigidly to each other to rotate together about a single axis is show in the diagram. The combined pulley is attached to a fixed stirrup and rotates about an axis passing through its centre of mass. The body of mass m is attached by a string to the smaller pulley, while the spring is attached by a string to the larger pulley. The stiffness of the spring is k, the moment of inertia of the combined pulley is l and the radius of each the larger and smaller pulley are r1 and r2 respectively.

Explanation / Answer

if the spring extends by 'x' distance then a point on the perifery of the bigger pulley rotates by the same amount i.e. 'x'

now,

arc length = x = r2 x theta......................1

but as the twp pulley are rigidly fixed together thus

theta remains same for both

but as radius is different the slack created in the string passing over smaller pulley will be

y = r1 x theta

putting theta from 1

theta = x/r2

y = r1x/r2

gain in spring potential enrgy = .5 k x ^2

loss in potential enrgy of the weight on the other end = mgy

and the difference will be stored in the pulley as kinetic enrgy as = .5 I w^2

where w = angular velocity of the pulley

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