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Question

Two particles are moving in different circles in same plane with different angular velocities as shown in figure. At t=0, initial positions of particles A and B are shown by dots on the respective circles. Initial distance between particles is 1 m. Particle A move anticlockwise in the first circle whereas B moves clockwise in the second circle. Angle described (rotated) by A and B in time t are θA=(π2t) and θB=(πt) respectively. Here θ is in radian and t is in second. Radius of each circle is shown in diagram. At time t=2 s, the angular velocity of the particle A with respect to the particle B is

A
5π6 rad/s
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B
3π2 rad/s
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C
π3 rad/s
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D
π2rad/s
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Solution

The correct option is A 5π6 rad/s
At time t=2 s, position of A and B are
θA=(π2t)=(π×22)=π rad
θB=(πt)=(π×2)=2π rad

vA=ω1r1=dθAdtr1=π2(1)=π2 m/sec
vB=ω2r2=dθBdtr2=2π m/sec

Distance AB=3 m
ω=vA+vBAB=π/2+2π3=5π6 rad/s

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