Figure shows two cylinders of radii r1 andr2 having moments of inertia I1 and I2 about their respective axes. Initially, the cylinders rotate about their axes with angular speeds ω1 and ω2 as shown in the figure. The cylinders are moved closer to touch each other keeping the axes parallel. The cylinders first slip over each other at the contact but the slipping finally ceases due to the friction between them. Find the angular speeds of the cylinder 1 and 2 respectively after the slipping ceases.
[I1ω1r2+I2ω2r1I2r21+I1r22]r2,[I1ω1r2+I2ω2r1I2r21+I1r22]r1
When slipping ceases, the linear speeds of the points of contact of the two cylinders will be equal. If ω′1 and ω′2 be the respective angular speeds, we have ω′1r1=ω′2r2...(i)
The change in the angular speed is brought about by the frictional force which acts as long as the slipping exists. If this force f acts for a time t, the torque on the first cylinder is fr1 and that on the second is fr2. Assuming ω1r1>ω2r2, the corresponding angular impulses are -fr1t and fr2t. We, therefore, have -fr1t=I1(ω′1−ω1)
fr1t=I1(ω′1−ω1)
and fr2t=I2(ω′2−ω2)
or,−I1r1(ω′1−ω1)I2r2(ω′2−ω2) ...(ii)
solving (i) and (ii),
ω′1=(I1ω1r2+I2ω2r1I2r21+I1r22)r2 and ω′2=(I1ω1r2+I2ω2r1I2r21+I1r22)r1.