Q.The position vector and angular velocity vector of a particle executing uniform circular motion at an instant are 2 i-cap and 4 k-cap respectively. Find its linear velocity at that instant.

Angular Velocity: The Language of Spinning

Imagine you're watching a ceiling fan. You know it's moving, but how do you describe how fast it's spinning? You could say "it makes 3 full turns every second" — that's a measure of angular velocity. But let's build this idea from the ground up.

The Intuition: Speed vs. Turning Speed

When a car moves in a straight line, we talk about its linear velocity — how many meters it covers per second. But when something rotates — a wheel, a planet, a spinning top — every point on it moves in a circle. The outer edge of a wheel travels a much longer distance in one rotation than a point near the centre. So if we tried to use ordinary speed (metres per second), we'd get different numbers for different parts of the same object. That's messy.

What we need is a quantity that describes the rotation itself, independent of how far a point is from the centre. That quantity is angular velocity.

The Core Idea

Angular velocity tells you how fast the angle is changing as something rotates. Instead of "metres per second," it's "radians per second" (or degrees per second, or revolutions per second).

The Precise Definition

Let an object rotate about a fixed axis. At time $t$, let its angular position be $\theta(t)$ — the angle it has turned through from some reference line. Then:

where $\omega$ (Greek letter omega) is the instantaneous angular velocity. For uniform rotation (constant speed), this simplifies to:

Units: radians per second (rad/s). In practice, you'll also see revolutions per minute (rpm) — 1 rpm = $\frac{2\pi}{60}$ rad/s.

Direction Matters: Angular Velocity as a Vector

Here's where it gets interesting. Angular velocity isn't just a number — it has a direction. But the direction isn't "clockwise" or "anticlockwise" in the plane of rotation. Instead, it points along the axis of rotation, following the right-hand rule:

So a spinning wheel's angular velocity vector points straight out from its axle. If the wheel spins faster, the vector gets longer. If it reverses direction, the vector flips.

Connecting to Linear Velocity

Here's the payoff: once you know the angular velocity of a rotating object, you can find the linear speed of any point on it. For a point at distance $r$ from the axis:

This is why the outer edge of a merry-go-round moves faster than a point near the centre — same $\omega$, different $r$.

A Concrete Example …