Moment Of Inertia, Perpendicular Axis Theorem, Important Topics For JEE 2025
Shrivastav 25 Jun, 2024
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Moment Of Inertia (I) : Like the centre of mass, the moment of inertia is a property of an object that is related to its mass distribution. The moment of inertia (denoted by I ) is an important quantity in the study of system of particles that are rotating. The role of the moment of inertia in the study of rotational motion is analogous to that of mass in the study of linear motion.
Moment of inertia gives a measurement of the resistance of a body to a change in its rotational motion. If a body is at rest, the larger the moment of inertia of a body, the more difficult it is to put that body into rotational motion. Similarly, the larger the moment of inertia of a body, the more difficult it is to stop its rotational motion. The moment of inertia is calculated about some axis (usually the rotational axis) and it depends on the mass as well as its distribution about that axis.Moment Of Inertia (I)
Moment of Inertia (MI) of a rigid n-particle system about an axis shown in Figure is the sum of the products of the masses of the particles with the square of their respective distances from the axis of rotation.
where m i is the mass of the i th particle and r i ⊥ is the perpendicular distance of the i th particle from the Axis of Rotation ( AOR ). It is a tensor quantity with SI unit kgm 2 .
Moment of Inertia depends upon the mass of the body and the manner the mass is distributed in the system with respect to the AOR . So, the moment of inertia depends on the location of the axis, that is, on how the mass of the body is distributed relative to the axis. Thus, a body does not possess a unique moment of inertia and different axes through the body are associated with different moments of inertia.
The role played by MI in rotational dynamics is actually the same as that played by inertia in linear dynamics. Moment of Inertia opposes any change in rotational motion and hence is also called rotational inertia. To find the Moment of Inertia of a continuous mass distribution, we consider an element of mass dm at a perpendicular distance r from the AOR as shown in Figure.
Perpendicular Axis Theorem
This theorem is distinct for a 2-D body (such as a lamina or a planar body) and a 3-D body (such as cube, prism etc).
For a Lamina: The theorem states that the moment of inertia of a lamina about an axis perpendicular to the plane of lamina (say z -axis) is equal to the sum of the moments of inertia of lamina about two mutually perpendicular axis lying in its own plane (say x and y -axis) intersecting each other at the point where the two perpendicular axis meet.
If I x and I y be the moment of inertia of lamina about two mutually perpendicular axis x and y lying in its plane and I z be the moment of inertia of lamina about third axis perpendicular to lamina and passing through intersection of x and y , then
I z = I x + I y
Parallel Axis Theorem
This theorem has the same statement for a planar (laminar) body as well as a three-dimensional body. It states that the moment of inertia of a body about any axis is equal to its moment of inertia about a parallel axis through its centre of gravity ( CG ) (or centre of mass ( CM )) plus the product of the mass of the body and the square of the perpendicular distance between the two parallel axes.
Mathematically,
Where, M is the total mass of the body, I G is the MI of body about CG and d is the perpendicular distance between two parallel axes.
Moment of Inertia of Different Bodies
| Body |
Axis of Rotation |
Diagram Showing Axis of Rotation |
Moment of Inertia |
| 1. Uniform thin bar | (a) Through centre of gravity and perpendicular to length. |
|
|
| (b) Through one end and perpendicular to length. |
|
|
|
| 2. Rectangular lamina | Passing through its C.G. and perpendicular to its plane of length and breath. |
|
|
| 3. Ring or Hoop | (a) Passing through its centre and perpendicular to its plane. |
|
MR 2 |
| (b) About diameter |
|
|
|
| 4. Disc | (a) Passing through its centre and perpendicular to its plane |
|
|
| (b) About diameter |
|
|
|
| 5. Hollow disc/Annular disc of radii R 1 and R 2 | Passing through its centre and perpendicular to its plane. |
|
|
| 6. Solid cylinder | (a) About its own geometric axis |
|
|
| (b) Passing through C.G. and perpendicular to its geometric axis. |
|
|
|
| 7. Hollow cylinder | (a) About its own geometrical axis |
|
MR 2 |
| (b) Passing through C.G. and perpendicular to length |
|
|
|
| 8. Thin spherical shell | (a) About diameter |
|
|
| (b) About tangent |
|
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| 9. Solid sphere | (a) About diameter |
|
|
| (b) About tangent |
|
|
|
| 10.Diatomic molecule | Passing through centre of gravity and perpendicular to bond length |
|
|
Examples Of Moment Of Inertia
1. Find the moment of inertia of a semi-circular annular half disc (of mass M , outer radius R 2 , inner radius R 1 ), about an axis passing through its centre and perpendicular to its plane as shown in Figure.
Ans. Consider an infinitestimal half ring of radius x , width dx and mass dm as shown in figure.
So,
Moment of inertia of this elemental half ring about the specified is given by
dl = x 2 dm
2. Three particles of masses 1g, 2g and 3g are kept at points (2cm, 0), (0, 6cm), (4cm, 3cm). Find moment of inertia of all three particles (in gm cm –2 ) about, (a) x -axis (b) y -axis (c) z -axis
Ans. (a) About x -axis
I x = I 1 + I 2 + I 3
Here r = perpendicular distance of the particle from x -axis
I
1
= (1)(0)
2
+ (2)(6)
2
+ (3)(3)
2
= 99 g-cm
2
(b) About y -axis
I
y
Here r = perpendicular distance of the particle from y -axis
I
y
= (1)(2)
2
+ (2)(0)
2
+ (3)(4)
2
= 52 g-cm
2
(c) About z -axis
I
z
Here, r = perpendicular distance of the particle from z -axis
I
z
= (1)(2)
2
+ (2)(6)
2
+ (3)(5)
2
= 151 g-cm
2
Moment Of Inertia FAQs
Q.1 : What is moment of inertia?
Q.2 : Is moment of inertia scalar or vector quantity?
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