Question

Three spools $A,B,C$ each having moment of inertia $I=M{R}^2/4$ are placed on rough ground and equal force $F$ is applied at positions as shown in the figures. Then

• A. frictional force on spool $A$ acts in forward direction
• B. frictional force on spool $A$ acts in backward direction
• C. frictional force on spool $B$ acts in forward direction
• D. frictional force on spool $B$ and $C$ acts in backward direction

Answer 1

Answer: (A), (D)

The correct options are
A frictional force on spool $A$ acts in forward direction
D frictional force on spool $B$ and $C$ acts in backward direction

For case C : Let there is no friction initially.

The point O' tends to slide in forward direction due to both translational as well as rotational motion.Thus the friction force acts in backward direction in case C.

But in case A and B, the point O tends to slide forwards due to translational motion and backwards due to rotational motion. So friction force will act in the direction opposite to the acceleration of point O.

Let say there is no friction initially.

Case A: $Ma=F⟹a=\frac{F}{M}$ forwards

Also, $\tau =I\alpha$

$Fr=\frac{M{R}^2}{4}\alpha$ ...............(1)

$F\times \frac{R}{3}=\frac{M{R}^2}{4}\alpha ⟹R\alpha =\frac{4F}{3M}$ backwards

Thus net acceleration of O, $a_O=R\alpha -a=\frac{F}{3M}$ backwards

Hence friction force acts in forward direction in case A.

For case B: $Ma=F⟹a=\frac{F}{M}$ forwards

from (1) $F\times \frac{R}{5}=\frac{M{R}^2}{4}\alpha ⟹R\alpha =\frac{4F}{5M}$ backwards

Thus $a_o=a-R\alpha =\frac{F}{5M}$ forwards

Hence friction force acts in backward direction in case B.