Answer:
[tex]I_{corner} =\frac{\rho _{ab}}{3}(a^2+b^2)[/tex]
Step-by-step explanation:
By applying the concept of calculus;
the moment of inertia of the lamina about one corner [tex]I_{corner}[/tex] is:
[tex]I_{corner} = \int\limits \int\limits_R (x^2+y^2) \rho d A \\ \\ I_{corner} = \int\limits^a_0\int\limits^b_0 \rho(x^2+y^2) dy dx[/tex]
where :
(a and b are the length and the breath of the rectangle respectively )
[tex]I_{corner} = \rho \int\limits^a_0 {x^2y}+ \frac{y^3}{3} |^ {^ b}_{_0} \, dx[/tex]
[tex]I_{corner} = \rho \int\limits^a_0 (bx^2 + \frac{b^3}{3})dx[/tex]
[tex]I_{corner} = \rho [\frac{bx^3}{3}+ \frac{b^3x}{3}]^ {^ a} _{_0}[/tex]
[tex]I_{corner} = \rho [\frac{a^3b}{3}+ \frac{ab^3}{3}][/tex]
[tex]I_{corner} =\frac{\rho _{ab}}{3}(a^2+b^2)[/tex]
Thus; the moment of inertia of the lamina about one corner is [tex]I_{corner} =\frac{\rho _{ab}}{3}(a^2+b^2)[/tex]
