Establish the relation between torque and moment of Inertia in rotational motion and define from it moment of Inertia.

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Accepted Answer

>

> [!TLDR]
> τ = dL/dt = Iα for a rigid body (constant I); this defines moment of inertia I = τ/α, the rotational analogue of mass.
>
> **Angular momentum and torque:** For a system of particles (or a rigid body) rotating about an axis, the angular momentum is
> $$L = I\omega$$
> where I is the moment of inertia about that axis and ω is the angular velocity.
>
> Torque is defined as the rate of change of angular momentum (the rotational analogue of Newton's second law, F = dp/dt):
> $$	au = \frac{dL}{dt}$$
>
> **Deriving τ = Iα:** For a rigid body, the moment of inertia I about a given axis is constant (the mass distribution relative to that axis doesn't change), so:
> $$	au = \frac{d(I\omega)}{dt} = I\frac{d\omega}{dt} = I\alpha$$
> where $\alpha = d\omega/dt$ is the angular acceleration.
>
> **Defining moment of inertia:** From this relation,
> $$I = \frac{	au}{\alpha}$$
> …

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.Establish the relation between torque and moment of Inertia in rotational motion and define from it 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.Establish the relation between torque and moment of Inertia in rotational motion and define from it moment of Inertia.

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