The center of mass of a thin rectangular plate (fig - x) with sides of length $a$ and $b$, whose mass per…
The center of mass of a thin rectangular plate (fig - x) with sides of length $a$ and $b$, whose mass per unit area $(\sigma)$ varies as $\sigma=\frac{\sigma_0 x}{a b}$ (where $\sigma_0$ is a constant), would be ________
$\left(\frac{2}{3} a, \frac{\mathrm{~b}}{2}\right)$
$\left(\frac{a}{2}, \frac{\mathrm{~b}}{2}\right)$
$\left(\frac{1}{3} a, \frac{\mathrm{~b}}{2}\right)$
$\left(\frac{2}{3} a, \frac{2}{3} b\right)$
Solution
$\begin{aligned} & d m=\sigma d A \\ & \begin{aligned} d & =\sigma(d x)(d y)=\frac{\sigma_0 x}{a b}(d x)(d y) \\ x_{c o m} & =\frac{\int x d m}{\int d m}=\frac{\int x \frac{\left(\sigma_0 x\right)}{a b}(d x)(d y)}{\int_0 \frac{\sigma_0}{a b}(d x)(d y)} \\ & =\frac{\int_0^a x^2 d x \int_0^b d y}{\int_0^b x d x \int_0^b d y}=\frac{2 a}{3} \\ y_{c o m} & =\frac{\int y d m}{\int d m}=\frac{\int y\left(\frac{\sigma_0 x}{a b}\right)(d x)(d y)}{\int \frac{\sigma_0 x}{a b}(d x)(d y)}\end{aligned}\end{aligned}$ $=\frac{\int_0^a x d x \int_0^b y d y}{\int_0^a x d x \int_0^b d y}=\frac{b}{2}$ i.e., $\vec{r}_{\mathrm{com}} \equiv\left(\frac{2 a}{3}, \frac{b}{2}\right)$
