Question

A thin uniform plate in the shape of an equilateral triangle (of mass $m$) of height $h$ performs small oscillations about the horizontal axis coinciding with one of its sides. lf the moment of inertia of the body is $I=\frac{m{h}^2}{n}$, find $n$.

Answer 1

Let y$=$ height of elemental strip of length dx
Its mass $=$(ydx)p
where p $=$ density per unit area
As the dotted line (axis x $=$ h) divides it into two subtriangle,
Total moment of inertia $=I=2{\int }_0^h\left(pydx\right)(h-x{)}^2$
$\Rightarrow I=2{\int }_0^{h}pxtan\alpha dx[h^2+x^2-2hx]$
$=2p\frac{l}{2h}\left[{\int }_0^h\right(h^{2}x+x^3-2h{x}^2\left)dx\right]$
$=\frac{pl}{h}[\frac{h^4}{2}+\frac{h^4}{4}-\frac{2h^4}{3}]$
$=pl{h}^3[\frac{1}{2}+\frac{1}{4}-\frac{2}{3}]$
$=pl{h}^3\left[\frac{6+3-8}{12}\right]$
$=\frac{pl{h}^3}{12}=\left(\frac{plh}{2}\right)\left(\frac{h^2}{6}\right)$

Mass of triangle $=\frac{1}{2}lhp$
$\Rightarrow I=\frac{m{h}^2}{6}$
$\Rightarrow n=6$