Question
A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first 2 s it rotates through an angle $\theta_{1}$ and in the next 2 s it rotates through an angle $\theta_{2}$. The ratio $\frac{\theta_{2}}{\theta_{1}}$ is $\_\_\_\_$ .
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Solution
Sol. [Key Concept: For uniform angular acceleration $\theta \propto \ t^{2}$: $\theta_{1} = \frac{1}{2}\alpha{(2)}^{2}$ and $\theta_{2} = \frac{1}{2}\alpha{(4)}^{2} - \theta_{1}$, giving the ratio.]
Let $\alpha$ be angular acceleration
In first 2 sec
$$\theta_{1} = \frac{1}{2} \cdot \alpha(2)^{2} = 2\alpha $$Angular covered in 4 sec
$$\theta^{'} = \frac{1}{2}(\alpha)(4)^{2} = 8\alpha $$Angle in next 2 sec
$${\theta_{2} = \theta^{'} - \theta_{1} = 8\alpha - 2\alpha }{\theta_{2} = 6\alpha }{\frac{\theta_{2}}{\theta_{1}} = \frac{6\alpha}{2\alpha} = 3 }$$
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