A Massive Bar Oscillating between Two Identical Rotating Support Rollers

A heavy uniform bar of mass rests on top of two identical rollers that are continuously and rapidly turned in opposite directions with angular speed . The centers of the rollers are at a distance apart. The coefficient of kinetic friction between the bar and the roller surfaces is , a constant independent of the relative speed of the two surfaces. The bar, held at rest with its center at distance from the midpoint of the two rollers, is released at . This Demonstration depicts the subsequent motion.

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The motion obeys the differential equation, ; is the position of the center of gravity from the midpoint of the two rollers at an instant .
The bar executes simple harmonic motion, , for all amplitudes .
Notation:
C.G. is the center of gravity.
and are the normal forces acting on the bar through the points of the contact with the left and right cylinders, .
and are the kinetic frictional forces at the left and right contact points on the bar, and .
Reference: D. Kleppner and R. J. Kolenkow, An Introduction to Mechanics (Special Indian Edition), India: Tata McGraw–Hill, 1999 problem 6.9.
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