Question

# A rigid body rotates about a fixed axis with variable angular speed $$\omega = A - Bt$$ , where $$A$$ and $$B$$ are constants . Find the angle through which it rotates before it comes to rest.

Solution

## $$\omega=A-Bt$$Time at which it comes to rest is-$$\omega=0\implies T=\dfrac{A}{B}$$$$\implies \dfrac{d\theta}{dt}=A-Bt$$$$\implies \int d\theta=\displaystyle \int_{0}^{T} (A-Bt)dt$$$$\implies \theta=\left[At-\dfrac{B}{2}t^2\right]_{0}^{T}$$$$\implies \theta=AT-\dfrac{BT^2}{2}$$$$\implies \theta=\dfrac{A^2}{B}-\dfrac{A^2}{2B}$$$$\implies \theta=\dfrac{A^2}{2B}$$Physics

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