91Ó°ÊÓ

The moment of inertia of a body is the property of measuring the rotational inertia of the body quantitatively. This resists the angular acceleration of the body in rotational motion. It is given by the following expression.

$\mathit{l}\mathbf{=}\underset{\mathbf{}}{\mathbf{∑}}\mathit{m}{\mathit{r}}^{\mathbf{2}}$

Here, m is the mass, and r is the distance.

For the continuous mass distribution body, the integration method is used to find the moment of inertia value.

$\mathit{l}\mathbf{=}\mathbf{∫}{\mathit{r}}^{\mathbf{2}}\mathit{d}\mathit{m}$

Here, dm is the mass of the infinitesimal element and r is the distance of the mass element from the axis of rotation.

In part (d), the thickness of the plate must be less than a. In the moment on inertia expression, distance r must be perpendicular to the chosen axis of rotation.

In part (d), the axis of rotation is along the side of the plate, a is perpendicular to the chosen axis of rotation whereas b is parallel to the axis of rotation.

The thickness t in this case must be taken as small as if t is not very small In comparison with a, the it is required to take this distance into consideration while calculating the moment of inertia value. Only thickness value can be neglected when it is very small compared to a.

In part (c), the axis of rotation is at the center of plate. Both the a ,b is perpendicular to the chosen axis whereas the thickness is parallel to the axis of rotation.

Therefore, the thickness value does not affect the moment of inertia value in calculations.

Hence, in part (c) the expression given for lapplies no matter how thick the plate is whereas in part (d) thickness must be small than the a value.