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

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

$$
t=0.5 \mathrm{s}
$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$
\omega_{B}=? ?
$$

$$
@  t=3s
$$

$$
d \omega=\alpha d t
$$

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

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

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

$$
@{t}=3 s
$$

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

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

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

$$
\omega_ B=31.7 rad/s
$$

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