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Question

A particle moves on xy plane. Its position vector at any time t is r={(2t)^i+(2t2)^j}m. The rate of change of θ at time t=2 second. (Where θ is the angle which its velocity vector makes with positive x-axis) is


A

217 rad/s

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B

114 rad/s

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C

47 rad/s

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D

65 rad/s

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Solution

The correct option is A

217 rad/s


r=2t^i+2t2^j
Differentiating w.r.t t to get velocity,
v=2^i+4t^j
Angle that velocity makes with x-axis is slope =tanθ=4t2=2t .....(1)

Now, we have to find dθdt at t=2s, differentiating equation (1) we get
sec2θdθdt=2
dθdt=2sec2θ
dθdt=21+tan2θ=21+4t2 We know sec2θ=1+tan2θ
at t=2
dθdt=21+16=217 rad/sec


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