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

ω=A Bt Time at which it comes to rest is ω=0 T=AB dθdt=A Bt ∫ dθ=∫_0^T (A Bt)dt θ=[At B2t^2]_0^T θ=AT BT^22 θ=A^2B A^22B ...

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Standard XII

Physics

Angular Velocity

A rigid body ...

Question

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

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Solution

$ω = A - B t$

Time at which it comes to rest is-

$ω = 0 ⟹ T = \frac{A}{B}$

$⟹ \frac{d θ}{d t} = A - B t$

$⟹ \int d θ = \int_{0}^{T} (A - B t) d t$

$⟹ θ = {[A t - \frac{B}{2} t^{2}]}_{0}^{2}$

$⟹ θ = A T - \frac{B T^{2}}{2}$

$⟹ θ = \frac{A^{2}}{B} - \frac{A^{2}}{2 B}$

$⟹ θ = \frac{A^{2}}{2 B}$

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