Question:
The moment of inertia of uniform semicircular disc of mass $M$ and radius $r$ about a line perpendicular to the plane of the disc through the centre is:
The moment of inertia of uniform semicircular disc of mass $M$ and radius $r$ about a line perpendicular to the plane of the disc through the centre is:
Updated On: Aug 1, 2022
- $\frac{1}{4}Mr^{2}$
- $\frac{2}{5}Mr^{2}$
- $Mr^{2}$
- $\frac{1}{2}Mr^{2}$
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The Correct Option is D
Solution and Explanation
The mass of complete (circular) disc is
The moment of inertia of disc about the given axis is
$I=\frac{2\,Mr^{2}}{2}
=Mr^{2}$
Let, the moment of inertia of semicircular disc is $I_{1}$ The disc may be assumed as combination of two semicircular parts.
Thus, $I_{1}=I-I_{1}$
$\therefore I_{1}=\frac{I}{2}=\frac{Mr^{2}}{2}$
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Concepts Used:
Moment of Inertia
Moment of inertia is defined as the quantity expressed by the body resisting angular acceleration which is the sum of the product of the mass of every particle with its square of a distance from the axis of rotation.
Moment of inertia mainly depends on the following three factors:
- The density of the material
- Shape and size of the body
- Axis of rotation
Formula:
In general form, the moment of inertia can be expressed as,
I = m × r²
Where,
I = Moment of inertia.
m = sum of the product of the mass.
r = distance from the axis of the rotation.
M¹ L² T° is the dimensional formula of the moment of inertia.
The equation for moment of inertia is given by,
I = I = ∑mi ri²
Methods to calculate Moment of Inertia:
To calculate the moment of inertia, we use two important theorems-
- Perpendicular axis theorem
- Parallel axis theorem