Define moment of inertia and radius of gyration. What is the physical significance of moment of inertia?

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> [!TLDR]
> Moment of inertia I = Σmᵢrᵢ² measures a body's rotational inertia about an axis; radius of gyration k = √(I/M) is the equivalent point-mass distance; physically, I plays the role mass plays in linear motion.
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> **Moment of inertia:** for a system of particles rotating about an axis, it is defined as
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> $I = \sum_i m_i r_i^2$
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> the sum of each particle's mass times the square of its perpendicular distance from the axis of rotation. For a continuous body, $I = \int r^2\,dm$.
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> **Radius of gyration (k):** the distance from the axis at which the entire mass of the body could be assumed to be concentrated so as to give the same moment of inertia as the actual body:
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> $I = Mk^2 \quad\Rightarrow\quad k = \sqrt{\dfrac{I}{M}}$
> …

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 and radius of gyration. What is the physical significance of moment of inertia?

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 and radius of gyration. What is the physical significance of moment of inertia?

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