Understanding Moment of Inertia: Trick to Learn Moment of Inertia

Moment of Inertia: The property of a body by virtue of which it opposes any change in its state of rotation about an axis.

Explanation: The moment of inertia is defined by the equation:

I=∑m_ir_i²

where m_i is the mass of each point in the object and r_i is the distance from the axis of rotation. It depends on both the mass of the object and how that mass is distributed relative to the axis of rotation.

The term “mass moment of inertia” is often used to explicitly emphasize that the moment of inertia is dependent on the mass distribution of an object.

Radius of Gyration: The radius of gyration of a body about an axis of rotation is defined as the radial distance of a point from the axis of rotation at which, if whole mass of the body is assumed to be concentrated, its moment of inertia about the given axis would be the same as with its actual distribution of mass.

$k = \sqrt{\frac{I}{m}}$

Theorem of Parallel Axes: The moment of inertia I of a body about any axis is equal to the sum of moment of inertia I_cm about a parallel axis through its centre of mass and the product of the mass of the body and the square of distance between the two axes, i.e

I=I_cm + Ma²

Theorem of Perpendicular Axes: The Perpendicular Axis Theorem states that for an object lying in a plane, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia about two perpendicular axes in the plane of the object.

Mathematically, it is expressed as:

I_z=I_x+I_y

Note: The Perpendicular Axis Theorem is only applicable to 2-dimensional shapes, whereas the Parallel Axis Theorem can be applied to 3-dimensional shapes as well.

Physical significance of moment of inertia:

The moment of inertia plays the same role in the rotational motion as mass plays in translational motion. Just as mass determines inertia in linear motion, moment of inertia determines how much torque is required to rotate or stop an object around a specific axis. It depends not only on the object’s mass but also on how that mass is distributed relative to the axis of rotation.

Types of Moment of Inertia and Their Definitions (Rarely asked)

Mass Moment of Inertia:
Quantifies a body’s resistance to rotational motion about an axis, based on its mass distribution.
Area Moment of Inertia:
Measures an area’s resistance to bending when subjected to a load, commonly used in beam analysis.

There are other types of moments of inertia, but they are not generally asked about.

Trick to learn moment of inertia of some shapes

Let mass of each object be M and its radius be R. Firstly, list the different shapes in the order as in the above table. Then, write MR² under each column. Multiply each MR² by 2 and then divide it by 2, 3, 4, and 5 in the series. Now, simplify them to get the moment of inertia of different shapes across an axis passing through its centre. Here, two shapes have the same moment of inertia because a hollow cylinder is made up of several hollow rings and a solid cylinder is made up of several solid disks. The length of the object has no role in the calculation of the moment of inertia. It can also be noted that the moment of inertia decreases as mass is more distributed near the axis.
Moreover, we can apply parallel axis theorem to calculate the moment of inertia of any shape along its edge. For example: Moment of inertia of a solid disk along its edge is I_edge= I_z+ MR² =>MR²/2 + MR²=3/2MR²

Hollow Ring	Thin Spherical Shell	Solid Disc	Solid Sphere
Hollow Cylinder		Solid Cylinder
$I = \frac{2}{2} MR^2$	$I = \frac{2}{3} MR^2$	$I = \frac{2}{4} MR^2$	$I = \frac{2}{5} MR^2$
$I = MR^2$	$I = \frac{2}{3} MR^2$	$I = \frac{1}{2} MR^2$	$I = \frac{2}{5} MR^2$

Understanding Moment of Inertia

Trick to learn moment of inertia of some shapes

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