Define angular velocity (omega). Derive V = r x omega.

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> [!TLDR]
> Angular velocity measures how fast the angular position changes with time; combining it with the position vector through a cross product gives the particle's linear velocity, V = ω × r.
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> **Angular velocity.** Consider a particle P moving on a circular path of radius r about a fixed axis through centre O. Let its angular position at time t be θ, measured from a fixed reference line. In a small time interval dt, the particle sweeps through a small angle dθ.
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> The angular velocity is defined as the time rate of change of angular displacement:
> $$\omega = \frac{d	heta}{dt}$$
> Its SI unit is rad s⁻¹. Angular velocity is a vector quantity — its direction is along the axis of rotation, given by the right-hand rule (curl the fingers of the right hand in the sense of rotation; the thumb points along **ω**).
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> **Derivation of V = r × ω (equivalently V = ω × r).**
> Let **r** be the position vector of the particle P drawn from a point O on the axis of rotation to P. As the particle rotates, in time dt it moves through an angle dθ and traces a small arc of length
> $$ds = r\,d	heta$$
> (since arc length = radius × angle, for small dθ).
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> Dividing both sides by dt:
> $$\frac{ds}{dt} = r\frac{d	heta}{dt}$$
> $$v = r\omega$$
> …

Q.Define angular velocity (omega). Derive V = r x omega.

Concept understanding — Angular Velocity

Angular Velocity: The Language of Spinning

The Intuition: Speed vs. Turning Speed

The Core Idea

The Precise Definition

Direction Matters: Angular Velocity as a Vector

Connecting to Linear Velocity

A Concrete Example …

Q.Define angular velocity (omega). Derive V = r x omega.

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