A flywheel with the initial angular velocity 12 rad / s decelerates due to the forces whose moment relative to the axis is proportional to the square root of its angular velocity. Find the mean angular velocity of the flywheel average over the total deceleration time in rads^-1

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Detailed Solution

According to the question,

Let $Iα = - k \sqrt{ω}$
$\begin{array}{l} I \frac{dω}{dt} = - k \sqrt{ω} \end{array}$

$I \frac{d ω}{\sqrt{ω}} = - kdt$

$Integrating, \sqrt{ω} = - \frac{kt}{2 I} + \sqrt{ω_{0}}$

$or, ω = \frac{k^{2} t^{2}}{4 I^{2}} - \frac{\sqrt{ω_{0} kt}}{I} + ω_{0}$

$(Noting that at t = 0, ω = ω_{0})$

$Let the flywheel stops at t = t_{0} then from equation (1), t_{0} = \frac{2 I \sqrt{ω_{0}}}{k}$

$<ω> = \frac{\int_{0}^{t_{0}} d d ωdt}{\int_{0}^{t_{0}} d dt} = \frac{ω_{0}}{3} = 4 rad / s$

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