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Standard XII

Physics

Angular Acceleration

The angular v...

Question

The angular velocity of a gear is controlled according to $ω = 12 - 3 t^{2}$ where $ω$ in radian per second, is positive in the clockwise sense and $t$ is the time in seconds. Find the net angular displacement $Δ θ$ from the time $t = 0$ to $t = 3 s .$ Also, find the number of revolutions $N$ through which the gear turns during the $3 s .$

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Solution

$θ = \int_{0}^{3} ω d t = \int_{0}^{3} (12 - 3 t^{2}) d t$
$- 36 - 27 = 9 r a d$
$N = \frac{θ}{2 π} = \frac{9}{2 \times 3.14} = 1.43$

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θ=∫30ωdt=∫30(12−3t2)dt−36−27=9radN=θ2π=92×3.14=1.43

$θ = \int_{0}^{3} ω d t = \int_{0}^{3} (12 - 3 t^{2}) d t$
$- 36 - 27 = 9 r a d$
$N = \frac{θ}{2 π} = \frac{9}{2 \times 3.14} = 1.43$