Define moment of inertia of a rigid body.

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Accepted Answer

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> [!TLDR]
> Moment of inertia $I = \sum m_i r_i^2$ measures how a body's mass is distributed relative to an axis, and thus its resistance to angular acceleration about that axis.
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> Just as mass measures a body's inertia (resistance to change in linear motion), **moment of inertia** measures a body's inertia in rotational motion — its reluctance to change its angular velocity when a torque is applied.
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> For a rigid body made up of particles of masses $m_1, m_2, m_3, \dots$ situated at perpendicular distances $r_1, r_2, r_3, \dots$ from a given axis of rotation, the moment of inertia about that axis is defined as:
> $$I = \sum_i m_i r_i^2 = m_1 r_1^2 + m_2 r_2^2 + \cdots$$
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> For a continuous body, this sum becomes an integral, $I = \int r^2\, dm$.
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> Key features: …

Linear Quantity	Rotational Analogue
Displacement $x$	Angular displacement $\theta$
Velocity $v$	Angular velocity $\omega$
Acceleration $a$	Angular acceleration $\alpha$
Mass $m$	Moment of inertia $I$
Force $F$	Torque $\tau$
Newton's 2nd law: $F = ma$	$\tau = I\alpha$
Kinetic energy: $\frac{1}{2}mv^2$	$\frac{1}{2}I\omega^2$
Momentum: $p = mv$	Angular momentum: $L = I\omega$

Q.Define moment of inertia of a rigid body.

Concept understanding — Rotational Dynamics

Rotational Dynamics: The Physics of Spinning Things

The Core Idea

Breaking It Down

Torque — The Rotational "Push"

Moment of Inertia — The Rotational "Mass"

Angular Acceleration — How Fast Rotation Changes

The Complete Picture: Rotational Analogues

The Key Insight: Conservation of Angular Momentum

Q.Define moment of inertia of a rigid body.

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