A Massive Bar Oscillating between Two Identical Rotating Support Rollers
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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.
Contributed by: Anindya Kumar Biswas (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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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