Ncert Exercises & Solutions

Q. 26 Two cylindrical hollow drums of radii R and 2R, and of a common height h, are rotating with angular velocities $ω$ (anti-clockwise) and $ω$ (clockwise), respectively. Their axes, fixed are parallel and in a horizontal plane separated by 3R+ $δ$ . They are now brought in contact ( $δ$ $\to$ 0)

(a) Show the frictional forces just after contact.

(b) Identify forces and torques external to the system just after contact.

Hint: No slipping(no friction) means there is no relative velocity between the two surfaces in contact.

Step 1: Show the frictional forces just after contact

(a) Consider the situation shown below, we have shown the frictional forces

Step 2: Identify forces and torques external to the system just after contact

(b)

$F' = F = F " where F' and F " are external forces through support .$
$\Rightarrow F_{net} = 0 (one each cylinder)$

External torque = F x 3R, (anti-clockwise)

Step 3: Find the ratio of angular velocities when friction ceases.

(c) Let $ω_{1} and ω_{2}$ be final angular velocities of smaller and bigger drum respectively (anti-clockwise and clockwise respectively).

Finally, there will be no friction.

$Hence, {Rω}_{1} = 2 {Rω}_{2} \Rightarrow \frac{ω_{1}}{ω_{2}} = 2$

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