A block of mass m is placed on a long trolley of mass M . Initially, the trolley is at rest on a smooth horizontal surface. A person of mass m p starts running from one end of the trolley with a con

Question

A block of mass m is placed on a long trolley of mass M . Initially, the trolley is at rest on a smooth horizontal surface. A person of mass m p ​ starts running from one end of the trolley with a constant velocity v 0 ​ relative to the trolley. If the person covers a distance L on the trolley, what is the displacement of the trolley on the ground?

TestHub · Accepted Answer

Since there are no external horizontal forces acting on the system (person + block + trolley), the center of mass of the system remains stationary.

Let $x_{p}$ , $x_{b}$ , and $x_{t}$ be the displacements of the person, block, and trolley, respectively, from their initial positions. The total mass of the system is $M_{t o t a l} = M + m + m_{p}$ . Since the center of mass does not move, $Δ X_{CM} = 0$ .

So, $m_{p} x_{p} + m x_{b} + M x_{t} = 0$ . The block is stationary relative to the trolley, so its displacement is the same as the trolley's, $x_{b} = x_{t}$ . The person moves a distance $L$ relative to the trolley. So, $x_{p} = x_{t} + L$ . Substituting these into the equation: $m_{p} (x_{t} + L) + m x_{t} + M x_{t} = 0$

$m_{p} L + (m_{p} + m + M) x_{t} = 0$ $x_{t} = - \frac{m _{p} L}{M + m + m _{p}}$ The negative sign indicates the trolley moves in the opposite direction to the person.

Physics - Center of Mass Question with Solution | TestHub

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