Question

A solid sphere of Mass $M$ and radius $R$ is pulled horizontally on a rough surface as shown in the figure. Choose the incorrect alternative(s).

• A. The magnitude of the frictional force is $F/3$.
• B. The frictional force on the sphere acts forward.
• C. The acceleration of the centre of mass is $2F/3M$.
• D. The acceleration of the centre of mass is $5F/7M$.

Answer 1

Answer: (A), (B), (C)

The correct options are
A The magnitude of the frictional force is $F/3$.
B The frictional force on the sphere acts forward.
C The acceleration of the centre of mass is $2F/3M$.


From Newton's law,
$F-f=M{a}_{cm}$ .....(1)
total torque equation $fR=I\times \alpha$ ....(2)
for pure rolling condition $a_{cm}=R\alpha$ .....(3)

for solid sphere moment of inertia =$\frac{2}{5}M{R}^2$
using eq. (2)
$f=\frac{2}{5}M{R}^2\times \frac{a_{cm}}{R^2}=\frac{2}{5}M{a}_{cm}M{a}_{cm}=\frac{5f}{2M}$

putting at eq. (1)
$(F-f)=\frac{5f}{2M}F=\frac{7f}{2}$
So, for the frication force, $f=\frac{2F}{7}$


The direction of fricational force is backward, but the option says it's in forward direction.

Hence, option A, B, is correct

Now calculation of angular acceleration,
$a_{cm}=\frac{f}{M}\Rightarrow \frac{\frac{2F}{7}}{M}\times \frac{5}{2}{a}_{cm}=\frac{5F}{7M}$

Hence, option C is correct and D is wrong.