Question

A plank of mass M is placed on a rough horizontal surface. A man of mass m walks on the plank with an acceleration 'a' while the plank is also acted upon by a horizontal force F whose magnitude and direction can be adjusted to keep the plank at rest. he coefficient of friction between plank and surface is $\mu$. Choose the correct options.

• A. The maximum value of F to keep plank at rest is ma$+\mu (M+m)g$
• B. The minimum value of F to keep the plank at rest is ma$-\mu (M+m)g$
• C. The direction of friction on ground due to plank will always be forward
• D. If $F=ma$ then friction on man due to plank is also F and friction between plank and ground will be zero

Answer 1

Answer: (A), (B), (D)

The correct options are
A The maximum value of F to keep plank at rest is ma$+\mu (M+m)g$
B The minimum value of F to keep the plank at rest is ma$-\mu (M+m)g$
D If $F=ma$ then friction on man due to plank is also F and friction between plank and ground will be zero

If man moves with an acceleration $a$, friction acts on man foot is $f_1$.

then,

$f_1=ma$

$f_2-$ frictional force between ground and plank

$f_{2max}=\mu (M+m)g$

If $F$ is maximum, then $f_{2max}$ act toward backward

$F_{max}=f_1+f_{2max}=ma+\mu (M+m)g$

If $F$ is minimum, then $f_{2max}$ act forward

$F_{min}=f_1-f_{2max}=ma-\mu (M+m)g$

If $F=ma=f$ is minimum, then $f_2=0$

As from above discussion, it is clear that $f_2$ direction. Hence options (A), (B), and (D) are correct.