Two identical rollers are mounted with their axes parallel, in a horizontal plan

Question

Two identical rollers are mounted with their axes parallel, in a horizontal plane, a
distance 2d = 39.2 cm apart. The two rollers are rotating inwardly at the top
with the same angular speed !. A long uniform board is laid across them in a direction
perpendicular to their axes. The board of mass m = 3:52 kg is originally placed so that
its center of mass lies a distance x0 = 10 cm from the point midway between the rollers.
The coecient of friction between the board and rollers is k = 0:653. What is the
period (s) of the motion?

Explanation / Answer

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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