Question

Derive an expression for moment of inertia of a rectangular plate of sides a and b, where a is the longer side; about an axis parallel to width b and passing through the centre.

what will be the moment of inertia about an axis perpendicular to the plane of the plate passing through its centre c?

Answer 1

(a) Moment of inertia of a rectangular plate of sides a and b, where a is the longer side; about an axis parallel to width b and passing through the centre-
Take a rectangular plate having length a and width b.Now draw a line AB passing through centre parallel to b.Draw a line parallel to AB at a distance x from it and another at a distance x+dx.We can take a strip enclosed between the two lines as the small element.
As plate is uniform,
Its mass per unit area = $\frac{M}{ba}$
Mass of the strip = $\frac{M}{ba}bdx=\frac{M}{a}dx$
The perpendicular distance of the strip from AB = x .
The moment of inertia of the strip about AB = dI = $\frac{M}{a}dx{x}^2$
$$\begin{gathered} Therefore, \\ I={\int }_{-a/2}^{a/2}\frac{M}{a}{x}^{2}dx=\frac{M{a}^2}{12} \end{gathered}$$

Moment of inertia about an axis perpendicular to the plane of the plate passing through its centre c-
According to the theorem of perpendicular axis, which is

Iz = Ix + Iy

here

Ix =$\frac{M{b}^2}{12}$

so,

Iy =$\frac{M{a}^2}{12}$

So,Iz = Ix + Iy
Iz = $$\begin{gathered} =\frac{M{a}^2}{12}+\frac{M{b}^2}{12} \\ =\frac{M(a^2+b^2)}{12} \end{gathered}$$