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Newton's Laws for System of Particles

Question

Two particles A and B are revolving with constant angular velocity on two concentric circles of radius $1 m$ and $2 m$ respectively as shown in figure. The positions of the particles at $t = 0$ are shown in figure. If $m_{A} = 2 k g$ , $m_{B} = 1 k g$ and $P_{A}$ and $P_{B}$ are linear momentum of the particles then what is the maximum value of $| P_{A} + P_{B} |$ in $k g m / s$ in subsequent motion of the two particles.

A

1

B

7

C

8

D

5

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Solution

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Since the angular velocities of both the masses are constant, their linear velocities would also remain constant using the relation $v = r ω$ .
Thus at each point their velocities change only in direction. Thus the momentum of particle A is $P_{A} = m_{A} v_{A}$ i.e. $2 \times 2 = 4 k g m / s$ and that of B is $P_{B} = m_{B} v_{B}$ i.e. $1 \times 3 = 3 k g m / s$ .
We will get their momentum as a maximum when the particles travel parallel to each other in the same direction.
Thus we get the maximum value of their momentum simply as the sum of their individual momentums i.e. $| P_{A} + P_{B} | = 4 - 3 = 1 k g m / s$ at all points.

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