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

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> 
> [!TLDR]
> Angular acceleration $\alpha = d\omega/dt$ is how fast the angular velocity changes; torque $	au = rF\sin	heta$ is the moment of a force. Adding up the torques on every particle of a rigid body gives $	au = I\alpha$, the rotational form of $F = ma$.
>
> **1. Angular acceleration ($\alpha$).** When a body rotates about a fixed axis, its angular velocity $\omega$ may change with time. The rate of change of angular velocity is called angular acceleration:
> $$\alpha = \frac{d\omega}{dt} = \frac{d^2	heta}{dt^2}$$
> Its SI unit is rad s$^{-2}$ and it is a vector directed along the axis of rotation.
>
> **2. Torque ($	au$).** The torque (moment of force) about an axis is the turning effect of a force. If a force $\vec{F}$ acts at a position $\vec{r}$ from the axis,
> $$\vec{	au} = \vec{r}	imes\vec{F}, \qquad 	au = rF\sin	heta$$
> where $	heta$ is the angle between $\vec{r}$ and $\vec{F}$. Its SI unit is N m.
>
> **3. Relation between torque and angular acceleration.** Consider a rigid body rotating about a fixed axis with angular acceleration $\alpha$. Take a particle of mass $m_i$ at perpendicular distance $r_i$ from the axis. Its linear (tangential) acceleration is
> $$a_i = r_i\,\alpha$$
> so the tangential force on it is $F_i = m_i a_i = m_i r_i \alpha$, and the torque of this force about the axis is …

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

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