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A wheel rotates with a constant angular acceleration of $2 rad s^{-2}$. If the wheel starts from rest, number of revolutions it makes in first 10 seconds will be approximately:
A. 32
B. 24
C. 16
D. 8

Last updated date: 13th Jun 2024
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Hint: When the wheel starts rotation with an acceleration, its angular velocity will increase with time. Hence in such cases, we have to use the equations of motion for circular motion. Further, one revolution means 2π radians. Hence we can convert the number of rotations into radians rotated by multiplying it by 2π and vice-versa.

Formula used: $\theta = \omega_{\circ}t + \dfrac 12 \alpha t^2$, where $\omega_{\circ }$ is the initial angular velocity of the body and $\alpha$ is the angular acceleration and $\theta$ is the angle ( in radian) rotated by the body in time ‘t’.

Complete step by step answer:
Given $\omega_{\circ }$= 0 as the body starts from rest.
t=10 sec and $\alpha$=$2 rad s^{-2}$
Putting the values in the equation: $\theta = \omega_{\circ}t + \dfrac 12 \alpha t^2$, we get
$\theta = (0)\times 10 + \dfrac 12 2 (10)^2 = 100 rad$
Hence the angle rotated by the body is 100 radians.
Now to calculate number of revolutions made by the body, dividing the angle by 2π, we get
$No.\ of\ revolutions\ = \dfrac{100}{2\pi} = 15.9$
Hence the body will rotate approx. 16 times.

So, the correct answer is “Option C”.

Note: One must take care of the units involved in the question. Sometimes angular velocity is given in terms of revolutions per second also. Chance of mistake here is high that in the majority of questions, we are supposed to calculate the angle (in radians or degree). But in some questions no revolutions are asked too (like this one). Hence one should know that units and conversions between various units like radian to revolution and radian to degree, etc.