Question

Determine the center of mass/centroid of the region.

Short answer

The center of mass or the centroid of the region is \((2,1)\).

Step-by-step solution

Given data

The given data is the region

Concept of Mass of lamina

The mass of the lamina from\(x = a\)to\(x = b\)is obtained as\(m = \rho \int_a^b f (x)dx\). The moment about\(x\)-axis is obtained as\({M_x} = \rho \int_a^b {\frac{1}{2}} {(f(x))^2}dx\)and the moment about\(y\)-axis is obtained as\({M_y} = \rho \int_a^b x f(x)dx\). Thus, the coordinates of the center of mass or centroid are\(\bar x = \frac{{{M_y}}}{m}\)and\(\bar y = \frac{{{M_x}}}{m}\).

Simplify the expression

From the given shape, the equation of the line inclined to \(x\)-axis is \(y = \frac{2}{3}x\) that holds the point (3,2).

Here, the value of \(x\) varies from 0 to 3.

Obtain the mass of the lamina \(m\) as follows.

\(\begin{aligned}m &= \rho \int_0^3 {\left( {\frac{2}{3}x} \right)} dx\\m &= \frac{{2\rho }}{3}\int_0^3 x dx\\m &= \frac{{2\rho }}{3}\left( {\frac{{{x^2}}}{2}} \right)_0^3\\m &= \frac{{2\rho }}{3}\left( {\frac{{{3^2}}}{2}} \right)\\m &= 3\rho \end{aligned}\)

Obtain the value of both the moment

Obtain the value of moment \({M_x}\) as follows.

\(\begin{aligned}{M_x} &= \rho \int_0^3 {\frac{1}{2}} {\left( {\frac{2}{3}x} \right)^2}dx\\{M_x} &= \frac{\rho }{3}\int_0^3 {{x^2}} dx\\{M_x} &= \frac{\rho }{3}\left( {\frac{{{x^3}}}{3}} \right)_0^3\\{M_x} &= \frac{\rho }{3}\left( {\frac{{{3^3}}}{3}} \right)\\{M_x} &= 3\rho \end{aligned}\)

Obtain the value of moment \({M_y}\) as follows.

\(\begin{aligned}{M_y} &= \rho \int_0^3 x \left( {\frac{2}{3}x} \right)dx\\{M_y} &= \frac{{2\rho }}{3}\left( {\frac{{{x^3}}}{3}} \right)_0^3\\{M_y} &= \frac{{2\rho }}{3}\left( {\frac{{{3^3}}}{3}} \right)\\{M_y} &= 6\rho \end{aligned}\)

Obtain the center of mass/ centroid as follows: \(\bar x = \frac{{6\rho }}{{3\rho }} = 2{\rm{ and }}\bar y = \frac{{3\rho }}{{3\rho }} = 1\)

Thus, the center of the mass is at the point \((2,1)\).

Therefore, the center of mass or the centroid of the region is \((2,1)\).