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# A track star runs a 400-m race on a 400-m circular track in 45 s. What is his angular velocity assuming a constant speed?

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\begin{align*} \omega&=\dfrac{d\theta}{dt} \\ &=\dfrac{\Delta \theta}{\Delta t} \\ &=\dfrac{2\pi}{45} \\ &=0.14 \text{ } \dfrac{\text{rad}}{\text{s}} \end{align*}

Since the perimeter of the circular track is $400\text{ m}$, and the distance covered by the runner is also $400\text{ m}$, this means that the runner covered precisely one circumference, hence the angle covered is $2\pi \text{ rad}$. We proceed to use the definition of angular speed to calculate $\omega$. Note that $d\theta$ and $dt$ are equal to $\Delta \theta$ and $\Delta t$ respectively, because the rotation occurs at constant speed.

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