Constant Angular Acceleration

The particle began to move from rest along a circle with a constant angular acceleration. After five cycles (n = 5), its angular velocity reached the value ω = 12 rad/s.
Calculate the magnitude of the angular acceleration ε of this motion and the time interval required for the first 5 cycles.

Correct answer:

ε =  2.2918 rad/s²
t =  5.236 s

Step-by-step explanation:

$$\begin{gathered} \omega =12\text{ rad/s } \\ n=5 \\ \alpha =n\cdot 2\pi =5\cdot 2\cdot 3.1416\doteq 31.4159\text{ rad } \\ \alpha ={\displaystyle \frac{1}{2}}\cdot \epsilon \cdot t^2 \\ \epsilon =\omega \mathrm{/}t \\ \alpha ={\displaystyle \frac{1}{2}}\cdot \omega \mathrm{/}t\cdot t^2 \\ \alpha ={\displaystyle \frac{1}{2}}\cdot \omega \cdot t \\ t=2\cdot \alpha \mathrm{/}\omega =2\cdot 31.4159\mathrm{/}12\doteq 5.236\text{ s } \\ \epsilon =\omega \mathrm{/}t=12\mathrm{/}5.236=2.2918{\text{rad/s}}^2 \end{gathered}$$

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