A wheel initially at rest, is rotated with a uniform angular acce… - Vidyadip

Question

A wheel initially at rest, is rotated with a uniform angular acceleration. The wheel rotates through an angle ${\theta _1}$ in first one second and through an additional angle ${\theta _2}$ in the next one second. The ratio $\frac{{{\theta _2}}}{{{\theta _1}}}$ is

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Answer

Angular displacement in first one second ${\theta _1} = \frac{1}{2}\alpha \,{(1)^2} = \frac{\alpha }{2}$ ......$(i)$ [From $\theta = {\omega _1}t + \frac{1}{2}\alpha \,{t^2}$]

Now again we will consider motion from the rest and angular displacement in total two seconds

${\theta _1} + {\theta _2} = \frac{1}{2}\alpha \,{(2)^2} = 2\alpha $ ......$(ii)$

Solving $(i)$ and $(ii)$, we get ${\theta _1} = \frac{\alpha }{2}$ and ${\theta _2} = \frac{{3\alpha }}{2}$ $\therefore $ $\frac{{{\theta _2}}}{{{\theta _1}}} = 3$.

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