$I_x\= \int {\int }_R\mathrm{\ρ} y^2 \mathrm{dA}$

$I_y\= \int {\int }_R\mathrm{\ρ} x^2 \mathrm{dA}$

Table 6.6.1   Moments of inertia

Angular position

$\mathrm{\θ}\left(t\right)$

Angular velocity

$\stackrel{\.}{\mathrm{\θ}}\left(t\right)$ 

Angular acceleration

$\stackrel{..}{\mathrm{\θ}}\left(t\right)$ 

Arc length: $s\left(t\right)$ 

$s\left(t\right)\=r \mathrm{\θ}\left(t\right)$ 

Rim (or linear) speed: $\stackrel{\.}{s}\left(t\right)$ 

$\stackrel{\.}{s}\left(t\right)\=r \stackrel{\.}{\mathrm{\θ}}\left(t\right)$

Linear acceleration on rim: $a\=\stackrel{..}{s}\left(t\right)$

$\stackrel{..}{s}\left(t\right)\=r \stackrel{..}{\mathrm{\θ}}\left(t\right)$ 

Table 6.6.2   Rotation of $\mathrm{dm}$ about $x$-axis

$F$

$\=m a$

$\mathrm{\τ}\=r F$

$\=r \left(m a\right)$

$\=r m \left(\stackrel{..}{s}\left(t\right)\right)$

$\=r m \left(r \stackrel{..}{\mathrm{\θ}}\left(t\right)\right)$

$\=m r^2 \stackrel{..}{\mathrm{\θ}}\left(t\right)$

Example 6.6.1

The lamina occupies $R$, the region bounded by $y\=\sqrt{1-x^2}$ and the $x$-axis.

See Example 6.5.1.

Example 6.6.2

The lamina has the shape of $R$ in Example 6.6.1, and its density is equal to the square of the distance from the point $\left(0\,1\/2\right)$. See Example 6.5.2.

Example 6.6.3

The lamina has the shape of the triangle whose vertices are $\left(0\,0\right)\,\left(a\,0\right)\,\left(b\,c\right)$, where $a\,b\,c$ are positive, and $a\>b$. See Example 6.5.3.

Example 6.6.4

The lamina has the shape of the triangle whose vertices are $\left(0 \,0\right)\,\left(0\,3\right)\,\left(4\,0\right)$, and whose density is $\mathrm{\ρ}\=1\+2 x\+ y\/2$. See Example 6.5.4.

Example 6.6.5

The lamina is a semicircle whose density is equal to the distance from the center of the circle, and whose radius is 1. See Example 6.5.5.

Example 6.6.6

The lamina has the shape of the cardioid $r\=1\+\cos \left(\mathrm{\θ}\right)$, with density $\mathrm{\ρ}\=\left(1\+2 x^2\+3 y^2\right)\/10$. See Example 6.5.6.

Example 6.6.7

The lamina has density $\mathrm{\ρ}\=\left(1\+2 x^2\+3 y^2\right)\/10$, and is in the shape of $R$, the region bounded by $x\=y^2$ and $y\=x-2$. See Example 6.5.7.

Example 6.6.8

The lamina has the shape of an isosceles right triangle with legs of length 1, and has density that is twice the square of the distance from the vertex of the right angle.

Hint: Put the right angle at the origin. See Example 6.5.8.

Example 6.6.9

The lamina has the shape of $R$, the region bounded by the graphs of $x\=0$, and $x\=y-y^3$, where $y\in \left[0\,1\right]$. See Example 6.5.9.

Example 6.6.10

The lamina with density $\mathrm{\ρ}\=3\+2 x\+3 y$ has the shape of $R$, the region bounded by the ellipse $x^2\+4 y^2\=1$. See Example 6.5.10.