# A pulley of radius 2 m is rotated about its axis by a force F =20t-5t2 newton

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A pulley of radius 2 m is rotated about its axis by a force F =20t-5tnewton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kgm2 , the number of rotations made by the pulley before its direction of motion it reversed,

Options:

(a) more than 6 but less than 9

(b) more than 9

(c) less than 3

(d) more than 3 but less than 6

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Torque is given by τ = FR

Or α = FR/I

Given

F =20t – 5t2 R = 2m,

I = 10 kgm2

α = $$(20t-5t^2)\times2\over 10$$

α = 4t – t 2

$$\omega=\int_t^2\alpha dt=2t^2-t^3/3$$

at $$\omega =0 =2t^2-t^3/3$$

$$t^3=6t^2$$

$$t=6$$

$$\theta=\int_0^6 \omega dt=\int_0^6(2t^2-t^3/3)dt$$

$$\theta={ 36\over2\pi} <6$$

Hence, your answer should be (d) more than 3 but less than 6