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$$
\begin{array}{l}{\text { 17-2. The solid cylinder has an outer radius } R \text { , height } h \text { . }} \\ {\text { and is mado from a material having a clonsity that varics }}\end{array}
$$

$$
\begin{array}{l}{\text { From its conter as } \rho=k+\text { ar }^{2} \text { . Where } k \text { and } a \text { are constants. }} \\ {\text { Determine the mass of the eylinder and its moment of }} \\ {\text { inertia about the } z \text { ax is. }}\end{array}
$$

$$
\rho=k+{a r}^{2}
$$

$$
\mathrm{shell}
$$

$$
r \rightarrow d m
$$

\(∵ dm=\rho d{v}=\rho(2 \pi r d r) h \)

\(∴ m=\int_{0}^{R}\left(k+a r^{2}\right)(2 \pi r d r) h \)

$$
=2 \pi h \int_{0}^{R}\left(k r+a r^{3}\right) d r = 2 \pi h\left(\frac{k R^{2}}{2}+\frac{a k^{4}}{4}\right) \
$$

$$
m=\pi h R^{2}\left(k+\frac{a K^{2}}{2}\right)
$$

\(∵ d I=r^{2} dm=r^{{2}} \rho(2 \pi r d r) h \)

\(∴I_{z}=\int_{0}^{R} r^{2}\left(k+a r^{2}\right)(2 \pi r d r) h=2 \pi h \int_{0}^{R}\left(k r^{3}+a r^{5}\right) d r \)

\(∴I_{z}=2 \pi h\left[\frac{k R^{4}}{4}+\frac{a R^{6}}{6}\right] \Rightarrow I_{z}= \frac{\pi h{R}^{4}}{2}\left[k+\frac{2 a R^{2}}{3}\right] \)

\( \begin{array}{l}{17-10 . \text { The pendulum consists of a } 4-\mathrm{kg} \text { circular plate and }} \\ {\text { a } 2-\mathrm{kg} \text { slender rod. Determine the radius of gyration of the }} \\ {\text { pendulum about an axis perpendicular to the page and }} \\ {\text { passing through point } \text{O} \text { . }}\end{array} \)

$$
m_{c}=4 k g
$$

$$
m_{r}=2 k_{2}
$$

$$
I_{0}=\Sigma(I_G+md^2
$$

 

 

 

 

$$
I_{0}=\left[\frac{1}{12} m l^{2}+m_{r} d_{1}^{2}\right]+\left[\frac{1}{2} ml^{2}+m_{c} d_{2}^{2}\right]
$$

$$
=\left[\frac{1}{12}(2)(2)^{2}+2(1)^{2}\right]+\left[\frac{1}{2}(4)(0.5)^{2}+4(2.5)^{2}\right]
$$

$$
=28.17 \mathrm{kg}.\mathrm{m}^{2}
$$

$$
k_{0}=\sqrt{\frac{I_0}{m_{t}}}=\sqrt{\frac{28.17}{4+2}}=2.167 \mathrm{m}=2.17m
$$

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