Question

A square plate of mass M and edge L is shown in figure. Find the moment of inertia of the plate about the axis in the plane of plate passing through one of its vertex making an angle ${15}^0$ from horizontal.

Answer 1

Step !: $x$ is parallel to given axis.

$o$ is the centre of square, so $AO$ is the half of its diagonal.

so, $AO=\frac{L}{\sqrt{2}}$

$AE$ is the perpendicular distance of $x$ axis from given axis.

So, $△AOE$ is a right angle triangle.

$AE=r=AO\cos {30}^o$

$r=\frac{L}{\sqrt{2}}.\frac{\sqrt{3}}{2}$

$=\frac{\sqrt{3}L}{2\sqrt{2}}$

FOR FIGURE 2:

$x$ axis and $y$ axis are perpendicular but are in same plane.$z$ axis is perpendicular to the plane of $x$ and $y$ axis

So, according to perpendicular axis theorem,

$I_x+I_y=I_z$

$\Rightarrow I_x=I_y=\frac{I_z}{2}$

now we know,

$I_z=\frac{{ML}^2}{6}$

So $I_x=I_y-\frac{M{L}^2}{12}$

$I^{\prime }=I_x+M{r}^2=\frac{M{L}^2}{12}+M{\left(\frac{\sqrt{3}L}{2\sqrt{2}}\right)}^2=\frac{M{L}^2}{12}+\frac{3L^2}{8}$

$\Rightarrow I^{\prime }=\frac{11M}{24}{L}^2$