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Question

A system of uniform cylinders is shown in figure. All the cylinders are identical and there is no slipping at any contact. Velocity of lower and upper plate is V and 2V respectively as shown in figure. Then, the ratio of angular speed of the upper cylinder to that of the lower cyllinder is


A
3
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B
1/3
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C
1
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D
none of these
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Solution

The correct option is A 3

Given, no slipping at any contact. Hence velocity of both (plate and cylinder) will be the same at contact points.

For upper cylinder: (Assume V1 and ω1 as the linear velocity and angular velocity respectively)
VP=VQ, VR=VS
VP=2V, VR=V
(Assume rightward movement as positive)


As we know, VP=VO1+VP,O1
2V=V1+ω1R ...(i)
VR=VO1+VR,O1
V=V1ω1R ...(ii)
From (i) and (ii)
ω1=3V2R

For lower cylinder: (Assume V2 and ω2 as the linear velocity and angular velocity respectively)
VA=VB, VC=VD (for no slipping)
VA=V, VC=0


From , VA=VO2+VA,O2
V=V2ω2R ...(iii)
VC=VO2+VC,O2
0=V2+ω2R ...(iv)
From Eq. (iii) and (iv)
ω2=V2R
Thus,
ω1ω2=3V/2RV/2R=31

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