Consider a system involving a man and a plank, and a rocket. Match the following quantities related to the motion of the center of mass (COM): List - I List - II (P) Displacement of a uniform plank of

Question

Consider a system involving a man and a plank, and a rocket. Match the following quantities related to the motion of the center of mass (COM):

List - I	List - II
(P) Displacement of a uniform plank of mass $M$ and length $L$ resting on a smooth horizontal surface, when a man of mass $m$ walks from one end to the other.	(1) $0$
(Q) Velocity of the COM of the man-plank system, if the man walks with constant speed $v_{m}$ relative to the plank, and the system was initially at rest.	(2) $\frac{F}{m + M}$
(R) Acceleration of the COM of the man-plank system if an external horizontal force $F$ acts on the plank.	(3) $v_{e} ln (\frac{M _{0}}{M _{0} - k t})$
(S) Velocity of the rocket at time $t$ , given initial mass $M_{0}$ , constant exhaust velocity $v_{e}$ relative to the rocket, and constant mass ejection rate $k$ .	(4) $- \frac{m L}{m + M}$

TestHub · Accepted Answer

P: No external horizontal force, so the COM of the man-plank system stays fixed. If the plank shifts $x_{p}$ , the man's displacement relative to ground is $L + x_{p}$ .

$m (0) + M (L /2) = m (L + x_{p}) + M (L /2 + x_{p})$

$0 = m L + (m + M) x_{p}$ ⇒ $x_{p} = - m L / (m + M)$ (P)-(4)

Q: No external horizontal force and the system starts at rest, so $v_{COM} = 0$ throughout. (Q)-(1)

R: $a_{COM} = F_{e x t} / M_{t o t a l} = F / (m + M)$ (R)-(2)

S: Rocket equation: $v (t) = v_{e} ln (M_{0} / M (t))$ with $M (t) = M_{0} - k t$

$v (t) = v_{e} ln (M_{0} / (M_{0} - k t))$ (S)-(3)

Physics - Center of Mass Question with Solution | TestHub

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