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A rod is rotating with an angular velocity of $3 π t^{3}$ rad/s. The angle rotated by the rod from t = 0 to the time when it has an angular velocity of $24 π$ rad/s is given by $α π$ radian. Find $\frac{α}{2}$ .

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answer is 6.

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Detailed Solution

$\begin{array}{l} ω = 3 π t^{3} = 24 π \\ t = {(8)}^{1 / 3} = 2 \sec \\ \int_{0}^{θ} d θ = \int_{0}^{2} ω d t \Rightarrow {[\frac{3 π t^{3}}{4}]}_{0}^{2} \\ θ = \frac{3 π \times 16}{4} = 12 π r a d . \end{array}$

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