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. $3$
• B. $1/3$
• C. $1$
• D. none of these

Answer 1

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 $V_1$ and ${\omega }_1$ as the linear velocity and angular velocity respectively)
$V_P=V_Q,V_R=V_S$
$\Rightarrow V_P=2V,V_R=-V$
(Assume rightward movement as positive)


As we know, $V_P=V_{O_1}+V_{P,O_1}$
$2V=V_1+{\omega }_{1}R...\left(i\right)$
$V_R=V_{O_1}+V_{R,O_1}$
$-V=V_1-{\omega }_{1}R...\left(ii\right)$
From (i) and (ii)
${\omega }_1=\frac{3V}{2R}$

For lower cylinder: (Assume $V_2$ and ${\omega }_2$ as the linear velocity and angular velocity respectively)
$V_A=V_B,V_C=V_D$ (for no slipping)
$V_A=-V,V_C=0$


From , $V_A=V_{O_2}+V_{A,O_2}$
$-V=-V_2-{\omega }_{2}R...\left(iii\right)$
$V_C=V_{O_2}+V_{C,O_2}$
$0=-V_2+{\omega }_{2}R...\left(iv\right)$
From Eq. (iii) and (iv)
${\omega }_2=\frac{V}{2R}$
Thus,
$\frac{{\omega }_1}{{\omega }_2}=\frac{3V/2R}{V/2R}=\frac{3}{1}$