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$\begin{array}{l}{\text { 16-1. The angular velocity of the disk is defined by }} \\ {\omega=\left(5 t^{2}+2\right) \text { rad/s, where } t \text { is in seconds. Determine the }} \\ {\text { magnitudes of the velocity and acceleration of point } A \text { on }} \\ {\text { the disk when } t=0.5 \mathrm{s} \text { . }}\end{array}$

$\omega=\left(5 t^{2}+2\right) rad / s$

$t=0.5 \mathrm{s}$

$∵ \omega=\left(5 t^{2}+2\right) \Rightarrow \alpha=\frac{d \omega}{d t}=10 t$

$@ t=0.5 s \Rightarrow \omega=3.25 \mathrm{rad} / \mathrm{s}, \ \alpha=5{rad}/s^{2}$
$v_{A}=\omega r=3.25 * 0.8 = 2.6 \mathrm{m} / \mathrm{s}$

$a_{t}=a r=5(0.8)=4 \mathrm{m} / \mathrm{s}^{2}$

$a_{n}=\omega^{2} r=(3.25)^{2}(0.8)=8.45 \mathrm{m} / \mathrm{s}^{2}$

$a_{A}=\sqrt{a_{t}^{2}+a_{n}^{2}}=\sqrt{(4)^{2} +(8 . 45)^{2}}=9. 35 \mathrm{m} / \mathrm{s}^{2}$

$\begin{array}{l}{\text { 16-6. A wheel has an initial clockwise angular velocity of }} \\ {10 \text { rad/s and a constant angular acceleration of } 3 \text { rad/s }^{2} \text { . }} \\ {\text { Determine the number of revolutions it must undergo to acquire }} \\ {\text { a clockwise angular velocity of } 15 \text { rad/s. What time is required? }}\end{array}$

$\omega_{0}=10\text { rad } / \mathrm{s}$

$\alpha_{c}=3 rad / s^{2}$

$\omega=15 \mathrm{rad} / \mathrm{S}$

$\omega^{2}=\omega_{0}^{2}+2 \alpha_{c}\left(\theta-\theta_{0}\right)$

$(15)^{2}=(10)^{2}+2(3)(\theta-0)$

$∴ \theta=\frac {20.83}{2 \pi} \mathrm{rad}=3.32 rev$

$\omega=\omega_{0}+\alpha_{c} t$

$15=10+3 t \Rightarrow t=1.67 S$

$\begin{array}{l}{16-18 . \text { A motor gives gear } A \text { an angular acceleration of }} \\ {\alpha_{A}=\left(2 t^{3}\right) \text { rad } / \mathrm{s}^{2}, \text { where } t \text { is in seconds. If this gear is }} \\ {\text { initially turning at } \omega_{A}=15 \mathrm{rad} / \mathrm{s}, \text { determine the angular }} \\ {\text { velocity of gear } B \text { when } t=3 \mathrm{s} \text { . }}\end{array}$

$\alpha_{A}=\left(2 t^{3}\right) \text { rad } / \mathrm{s}^{2}$

$\omega_{0 A}=1 5 \mathrm{rad} / \mathrm{s}$

$\omega_{B}=? ?$

$@ t=3s$

$d \omega=\alpha d t$

$\int_{15}^{\omega} d \omega=\int_{0}^{t} 2 t^{3} d t$

$\omega_{A}-15=\frac{1}{2} t^{4}]_{0}^{t}$

$\omega_{A}=[\frac{1}{2}t^{4} +15] \text { rad /s } \Rightarrow \frac{1}{2}(3)^{4}+15$

$@{t}=3 s$

$\omega_{ A}=55.5 rad / s$

$V_{A}=V_ B \Rightarrow \omega_{B} r_{B}=\omega_{A} r_{A}$

$\omega _B(175)=55.5(100)$

$\omega_ B=31.7 rad/s$