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A wheel is rotating at the rate of 33 revolution / min. if it comes to a stop in 20 seconds. Then the angular retardation will be:A) 0B) 11$\pi$ rad/${s^2}$C) $\dfrac{\pi }{{200}}\,\,{\text{rad/}}{{\text{s}}^2}$D) $\dfrac{{11\pi }}{{200}}\,\,{\text{rad/}}{{\text{s}}^2}$

Last updated date: 09th Apr 2024
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Hint Since the wheel is going to stop, hence final angular acceleration = 0. Now use the equation ${\omega _f} = {\omega _i} + \alpha t$ to get the value of α. Also convert units of ω to rad/sec.

Given that,
The Initial angular velocity of the wheel, (${\omega _i}$ )= 33 rev/min
The Final angular velocity of the wheel, (${\omega _f}$) = 0
Time taken by wheel to stop (t) = 20 second = 20s .
We know that,
The initial angular velocity in rad/s (${\omega _i}$) is given by =
$\Rightarrow 33 \times \dfrac{{2\pi }}{{60}} \\ \Rightarrow 1.1\pi \,{\text{rad/s}} \\$ ($\dfrac{{2\pi }}{{60}}$ is the conversion factor from revolution/minute to radian/sec.)
The angular retardation can be calculated by using the formula,
(Since retardation is mentioned, hence we have to take the negative value of alpha and not positive value.) .
${\omega _f} = {\omega _i} - \alpha t \\ \Rightarrow 0 = 1.1\pi - \alpha \times 20 \\ \Rightarrow \alpha = \dfrac{{1.1\pi }}{{20}} = \dfrac{{11\pi }}{{200}} \\$ (Putting the values given in the question, we have the following.)

Hence, the angular retardation of the wheel will be $\dfrac{{11\pi }}{{200}}$ rad/${s^2}$.

Note All the three equations of motion can be used in circular motion just by changing the few notations like “a” (acceleration) with $\alpha$(angular acceleration), S (displacement) with θ (angular displacement), velocity (v) with omega (ω). But time remains the same.
E.g.
$v = u + at$ is similar to ${\omega _f} = {\omega _i} + \alpha t$ .