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$\begin{array}{l}{\text { A bicycle wheel has an initial angular velocity of } 1.50 \mathrm{rad} / \mathrm{s} \text { . }} \\ {\text { (a) If its angular acceleration is constant and equal to } 0.300 \mathrm{rad} / \mathrm{s}^{2} \text { , }} \\ {\text { what is its angular velocity at } t=2.50 \mathrm{s} ?(\text { b) Through what angle }} \\ {\text { has the wheel turned between } t=0 \text { and } t=2.50 \mathrm{s} ?}\end{array}$

$w_{0z}=1.5 \mathrm{rad} / \mathrm{s}$

$\alpha_{z}=0.3 \text { rad/s }^{2}$

$t=2.5s$

$w_{z} = w_{0 z}+\alpha_{z} t$

$=1.5+0.3(2.5)=2.25 \mathrm{r}\mathrm{ad} / \mathrm{s}$

$\theta-\theta_{0}=w_{0 z} t+\frac{1}{2} \alpha_{z} t^{2}$

$=1.5(2.5)+\frac{1}{2}(0.3)(2.5)^{2}=4.69 \mathrm{rad}$

$\theta - \theta_{0}=\left(\frac{w_{0z}+w_{z}}{2}\right) t=\left(\frac{1.5+2.25}{2}\right)(2.5)=4.69 \mathrm{rad}$

$\begin{array}{l}{\text { A high-speed flywheel in a motor is spinning at } 500 \text { rpm }} \\ {\text { when a power failure suddenly occurs. The flywheel has mass } 40.0 \mathrm{kg}} \\ {\text { and diameter } 75.0 \mathrm{cm} . \text { The power is off for } 30.0 \mathrm{s} \text { , and during this }} \\ {\text { time the flywheel slows due to friction in its axle bearings. During }} \\ {\text { the time the power is off, the flywheel makes } 200 \text { complete revolu- }} \\ {\text { tions. (a) At what rate is the flywheel spinning when the power }}\end{array}$

$\begin{array}{l}{\text { comes back on? (b) How long after the beginning of the power }} \\ {\text { failure would it have taken the flywheel to stop if the power had }} \\ {\text { not come back on, and how many revolutions would the wheel }} \\ {\text { have made during this time? }}\end{array}$

$\theta-\theta_{0}=200 \mathrm{rev}$

$w_{0z}=500 \mathrm{rev} / \mathrm{min} \longrightarrow 8.333 \mathrm{rev} / \mathrm{s}$

$t=30 s$

$w_{z} \ ??$

$\rightarrow \theta - \theta_{0}=(\frac{w_{0z}+w_{z}}{2})t$

$2 * \frac{w_{0z}+w_ {z}}{2}=\frac{\theta-\theta_{0}}{t}$

$w_{0z}+w_ z=2\left(\frac{\theta-\theta_0}{t}\right)$

$w_{z}=2\left(\frac{\theta-\theta_{0}}{t}\right)-w_{0z}=2\left(\frac{200}{30}\right)-500$

$=5 \ \mathrm{rev} / \mathrm{s}=300 \ \mathrm{rpm}$

$\alpha_{z} \text {? ? } \rightarrow w_{z}=w_{0z}+\alpha_{z}t \rightarrow \alpha_{z}=\frac{w_{z}-w_{0 z}}{t}$

$\alpha_{z}=\frac{300-500}{30}=-0.1111 \mathrm{rev} / \mathrm{s}^{2}$

$w_{z}=0, \quad \alpha_{z}=-0.1111 \quad \text { rev } / \mathrm{s}^{2}$

$w_{0z}=8.333 \text { rev/s }$

$w_{z}=w_{0z}+\alpha_{z} t \rightarrow t=75 \mathrm{s}$

${\longrightarrow \theta-\theta_{0}=\left(\frac{w_{0z}+w_{z}}{2}\right) t=\left(\frac{8.333-0}{2}\right)(75)=312 \ rev}$

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