Question

A man of mass \[M\] stands at one end of the plank of length $L$ which lies at rest on a frictionless horizontal surface. The man walks to the other end of the plank. If the mass of the plank is $(M/3)$, the distance that the man moves relative to ground is?
A. $L$
B. $L/4$
C. $3L/4$
D. $L/3$

Answer 1

Hint: We can solve this type of problem with the help of the concept of centre of mass and the conservation of momentum. Centre of mass is an imaginary point where the whole mass of an object may be concentrated.

Complete answer:
Centre of mass may be present either on the body or outside the body for example disc, centre of mass presents on its centre and for the ring, centre of mass is present outside the centre. Let takes two mass ${m_1}$ and ${m_2}$ of particles, the centre of mass of the system is, ${X_{cm}} = \dfrac{{{m_1}{x_1} + {m_2}{x_2}}}{{{m_1} + {m_2}}}$
Where ${x_1}$ and ${x_2}$.

In our question, A man of mass \[M\] stands at one end of the plank of length $L$ which lies at rest on a frictionless horizontal surface and the mass of plank $M/3$, then the centre of mass of the system,
 $\dfrac{{M/3 \times L/2 + M \times 0}}{{M/3 + M}} = \dfrac{L}{8}$, Centre of mass of the system is $L$/8 distance away from the man,

 The man walks to the other end of the plank. The centre of mass of the system does not move, it remains at the same positions because there is no external force on the system.

Let $x$ be the displacement of the plank when the man walks to the other side, $\left( {L - x} \right)$ is the displacement of the man. At the starting the, centre of the mass is at the $L/8$. Since there is no external force then,

$\dfrac{L}{8} = \dfrac{{m\left( {L - x} \right) + \dfrac{m}{3}\left( {\dfrac{L}{2} - x} \right)}}{{m + \dfrac{m}{3}}}$, $x = \dfrac{{3L}}{4}$
Then $L - x = \dfrac{L}{4}$ , the distance that the man moves relative to ground .

So, the correct answer is “Option B”.

Note:
We know that when the external force is zero the body maintains its position. So if the centre of mass is moving with some constant velocity, it will be moving with the same velocity and the centre of mass also has the same velocity.