Define angular acceleration and torque. Establish the relation between angular acceleration and torque.

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>

> [!TLDR]
> Angular acceleration is the rate of change of angular velocity, torque is the rotational analogue of force, and they are related by $	au = I\alpha$ — the rotational counterpart of $F = ma$.
>
> **Definition of angular acceleration**
> Angular acceleration is the time rate of change of angular velocity of a rotating body:
> $$\alpha = \frac{d\omega}{dt}$$
> Its SI unit is $	ext{rad s}^{-2}$. It is a vector along the axis of rotation.
>
> **Definition of torque**
> Torque (or moment of force) is a measure of the turning (rotational) effect produced by a force about a given axis or point. If a force $\vec{F}$ acts at a position $\vec{r}$ relative to the axis, the torque is:
> $$\vec{	au} = \vec{r} 	imes \vec{F}$$
> Its magnitude is $	au = rF\sin	heta$, where $	heta$ is the angle between $\vec{r}$ and $\vec{F}$; its SI unit is newton-metre (Nm).
>
> **Relation between angular acceleration and torque**
> Consider a rigid body made up of many particles rotating about a fixed axis with angular acceleration $\alpha$. Consider one particle of mass $m_i$ at perpendicular distance $r_i$ from the axis. Its linear (tangential) acceleration is $a_i = r_i\alpha$. The tangential force needed to produce this is $F_i = m_i a_i = m_i r_i \alpha$.
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> The torque due to this force about the axis is:
> $$	au_i = r_i F_i = m_i r_i^2 \alpha$$
> …

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.Define angular acceleration and torque. Establish the relation between angular acceleration and torque.

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 angular acceleration and torque. Establish the relation between angular acceleration and torque.

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