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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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