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# A wheel has a constant angular acceleration of 5.0 rad/s². Starting from rest, it turns through 300 rad. (a) What is its final angular velocity? (b) How much time elapses while it turns through the 300 radians?

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From $\textbf{ the kinematics of the rotational motion }$ we know that :

$\omega_{f}^2 = \omega_{i}^2 +2\alpha(\theta_{f} - \theta_{i} )$

Where:

• $\omega_{f}$ is the final angular velocity of the body.
• $\omega_{i}$ is the initial angular velocity of the body.
• $\alpha$is the angular acceleration of the body.
• $\theta_{i}$ is the initial angular displacement .
• $\theta_{f}$ is the final angular displacement .
• $t$ is the time .

From $\textbf{givens}$ we know that : $\alpha = 5 \ \mathrm{rad/s^2}$ , $(\theta_{f} - \theta_{i}) = 300$ rad , $\omega_{i} = 0$ rad/s because the it begins from rest .

$\textbf{plugging}$ known information into the eq. of angular velocity .

\begin{align*} \omega_{f}^2& = \omega_{i}^2 +2\alpha(\theta_{f} - \theta_{i} )\\\\ \omega_{f} &= \sqrt{\omega_{i}^2 + 2 \alpha (\theta_{f} - \theta_{i})}\\\\ &=\sqrt{0 + 2 \times 5 \times 300 }\\\\ &=\sqrt{3000}\\\\ &=54.7722 \end{align*}

$\boxed{\omega_{f} = 54.7722 \ \; \mathrm{rad/s}}$

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