a boy of mass 'm' stands on one end of a wooden plank of length L and mass M. The plank is floating on water. If the boy walks from one end of the plank to the other end at a constant speed the resulting displacement of the plank is?
mL/(M+m)
To determine the resulting displacement of the plank when the boy walks from one end to the other, we need to consider the conservation of momentum.
When the boy starts walking, he exerts a force on the plank in one direction. According to Newton's third law of motion, the plank exerts an equal and opposite force on the boy.
Assuming there is no friction between the plank and the water, the only external force acting on the system is the force exerted by the boy on the plank.
Since there are no other external forces, the total momentum of the system (boy + plank) remains constant.
Initially, the system is at rest, so the total momentum is zero. As the boy starts walking, he begins to move in one direction, and the plank moves in the opposite direction to conserve the total momentum.
The resulting displacement of the plank can be calculated using the principle of conservation of momentum:
𝑚𝑖𝑑𝑖 = −𝑀𝑑
Where:
𝑚 = mass of the boy
𝑀 = mass of the plank
𝑑𝑖 = initial displacement of the boy
𝑑 = resulting displacement of the plank
Because the boy starts at one end of the plank and walks to the other end, the initial displacement of the boy (𝑑𝑖) is equal to the length of the plank (𝐿).
Substituting the values into the equation, we get:
𝑚𝐿 = −𝑀𝑑
To find 𝑑, we rearrange the equation:
𝑑 = (𝑚𝐿) / −𝑀
Therefore, the resulting displacement of the plank when the boy walks from one end to the other is given by (𝑚𝐿) / −𝑀. The negative sign indicates that the plank moves in the opposite direction of the boy's displacement.