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$\begin{array}{l}{\text { A uniform disk with mass } 40.0 \mathrm{kg} \text { and radius } 0.200 \mathrm{m}} \\ {\text { is pivoted at its center about a horizontal, frictionless axle that is }} \\ {\text { stationary. The disk is initially at rest, and then a constant force }} \\ {F=30.0 \mathrm{N} \text { is applied tangent to the rim of the disk. (a) What is }} \\ {\text { the magnitude } v \text { of the tangential velocity of a point on the rim of }}\end{array}$

$\begin{array}{l}{\text { the disk after the disk has turned through } 0.200 \text { revolution? (b) What }} \\ {\text { is the magnitude } a \text { of the resultant acceleration of a point on the }} \\ {\text { rim of the disk after the disk has turned through } 0.200 \text { revolution? }}\end{array}$

$\sum\tau_{z}=I \alpha_{z}$

${w_{z}}^{2}={w_{0 z}}^{2}+2 \alpha_{z}\left(\theta-\theta_{0}\right)$

${I=\frac{1}{2} M R^{2}=\frac{1}{2}(40)(0.2)^{2}=0.8 \mathrm{kg} \cdot \mathrm{m}^{2}}$

${F.r=I \alpha_{z} \longrightarrow \alpha_{z}=\frac{F.{r}}{I}=\frac{30 * 0.2 m}{0.8}}=7.5rad/s^2$

$\omega_{z}=\sqrt{2 \alpha_ z\left(\theta-\theta_{0}\right)}$

$=\sqrt{2 * 7.5(0.2)}=4.342 \mathrm{rad} / \mathrm{s}$

$V=r\cdot \omega=0.2 *4.342=0.868 \mathrm{m/s}$

$a=\sqrt{{a _{tan}}^{2}+{a_{rad}}^{2}}$

$a_{tan}=r \alpha=0.2 \times 7.5=1.5 \mathrm{m} / \mathrm{s}^{2}$

$a_\text { rad }=r \omega^{2}=0.2 *4.342^{2}=3.771 \mathrm{m} / \mathrm{s}^ 2$

$∴ \quad a=\sqrt{1.5^{2}+3.771^2}=4 \cdot 06 \mathrm{m} / \mathrm{s}^{2}$

$\begin{array}{l}{\text { A } 2.00-\mathrm{kg} \text { textbook rests on a frictionless, horizontal }} \\ {\text { surface. A cord attached to the book passes over a pulley whose }} \\ {\text { diameter is } 0.150 \mathrm{m} \text { , to a hanging book with mass } 3.00 \mathrm{kg} \text { . The sys- }} \\ {\text { tem is released from rest, and the books are observed to move } 1.20 \mathrm{m}} \\ {\text { in } 0.800 \mathrm{s} \text { . (a) What is the tension in each part of the cord? ( b) What }} \\ {\text { is the moment of inertia of the pulley about its rotation axis? }}\end{array}$

$m_{b_{1}}=2 k g \ , \quad m_{b_{2}}=3 k g$

$D=0.15 \rightarrow R=\frac{0.15}{2}$

$T_{1}=? ?$

$T_{2}=? ?$

$x-x_{0}=v_{0x} t+\frac{1}{2} a_{x} t^{2} \rightarrow a_{x}=\frac{a\left(x-x_{0}\right)}{t^{2}}=\frac{2(1.2)}{(0.8)^2}$

$=3.75 \mathrm{m/s}^{2}$

$∴ T_{1}=m_1 a_{1}=2(3.75)=7.5 N$

$T_{2}=m_{2}\left(9-a_{1}\right)=3(9.8-3.75)=18.2 \mathrm{N}$

$\left(T_{2}-T_{1}\right) \cdot R=(18. 2-7.5) \cdot \frac{0.15}{2}=0.803 \mathrm{N} \cdot \mathrm{m}$

$\tau=F. r$

$I=\frac{\tau}{\alpha} \rightarrow \alpha=\frac{a_{1}}{R}=50 \mathrm{rad} / \mathrm{s} ^2$

$∴ \rightarrow I=\frac{\tau}{\alpha}=0.016 \mathrm{kg} \cdot \mathrm{m}^ 2$

$\begin{array}{l}{\text { A wheel rotates without friction about a stationary hori- }} \\ {\text { zontal axis at the center of the wheel. A constant tangential force }} \\ {\text { equal to } 80.0 \mathrm{N} \text { is applied to the rim of the wheel. The wheel has }} \\ {\text { radius } 0.120 \mathrm{m} \text { . Starting from rest, the wheel has an angular speed }} \\ {\text { of } 12.0 \mathrm{rev} / \mathrm{s} \text { after } 2.00 \mathrm{s} \text { . What is the moment of inertia of the }} \\ {\text { wheel? }}\end{array}$

$\omega_{0}=0$

$\omega_{z}=12 \text { rev/s }=75.4 \mathrm{rad} / \mathrm{s}$

$t=2 s$

$\omega_{z}=\omega_{0z}+\alpha_{z}(t) \longrightarrow\alpha_{z}=\frac{\omega z}{t}=\frac{75.4}{2}=37.7 rad/s$

$\sum \tau_{z}=I \alpha_{z} \rightarrow I=\frac{\sum \tau_{z}}{\alpha_{z}}=\frac{F.{r}}{\alpha_{z}}$

$∴ I=\frac{80 * 0.12}{37.7}=0.255 \mathrm{kg}. \mathrm{m}^{2}$