Moment of Inertia

Point

The moment of inertia of a point particle is $\boxed{mr^2}$. $r$ is the distance from the axis of rotation. For future reference, it is important to note that moments of inertia always add.

Rod rotated about End

Because moments of inertia always add, we can think of a road as a lot of individual points, so we create this integral. So, the moment of inertia of one small point is $$\mathrm{d}I = r^2 \mathrm{d}m$$ $$\mathrm{d}m = \rho (\text{density}) * \mathrm{d}x = \dfrac{M}{L}\mathrm{d}x$$ Now, we can express $dI$. $$\mathrm{d}I = r^2 \dfrac{M}{L}dx$$ $$\int_0^L \dfrac{M}{L}r^2\, \mathrm{d}x = \dfrac{M}{L}\dfrac{L^3}{3} = \boxed{\dfrac{ML^2}{3}}$$

Rod rotated about Center

We keep coming back to one principle which is that moment of inertias add. The moment of inertia of a rotated rod about its center equals 2 * (rod cut in half about its end) That's just what we do! The length is cut in half, and because the density of the rod doesn't change, the mass is halved as well. $$2 * \dfrac{\left(\dfrac{M}{2}\right)\left(\dfrac{L}{2}\right)^2}{3} = \boxed{\dfrac{ML^2}{12}}$$

Solid Disk

Solid Cone

The important thing to remember is that the total inertia is the sum of many moment of inertias of solid discs. $$dI = \dfrac{1}{2}dm*r^2$$ Why is this true? Because the total moment of inertia is the sum of moment of inertias of solid disc, and the formula is $\frac{1}{2}mr^2$. $$dm = \rho \text{ (density) } * \pi{}r^2 \text{ (area) } \mathrm{d}x \text{ (width) }$$ $$\rho = \dfrac{M}{\pi{}R^2h\frac{1}{3}} = \dfrac{3M}{\pi{}R^2h}$$ $$\mathrm{d}m = \dfrac{3M}{hR^2}*r^2\mathrm{d}x$$ $$\int_0^h \frac{1}{2}\left(\dfrac{3M}{hR^2}*r^2\right)* r^2\mathrm{d}x$$ $r$ is equal to $y$, which is equal to $\frac{R}{h}x$ $$\dfrac{3M}{2hR^2}\int_0^h y^4 \,\mathrm{d}x$$ Looking at the picture above, $y = \dfrac{R}{h}x$, so the integral is $$\dfrac{3M}{2hR^2}\int_0^h \frac{R^4}{h^4}x^2 \,\mathrm{d}x$$ $$\dfrac{3MR^2}{2h^5}\int_0^h x^4 \,\mathrm{d}x$$ $$\dfrac{3MR^2}{2h^5} * \frac{h^5}{5} = \boxed{\frac{3MR^2}{10}}$$