For a rigid body, the angular velocity of any particle about a given axis of rotation is:

Correct — B. In a rigid body the distance between any two particles never changes, so every particle turns through the same angle in the same time.

Angular velocity is that angle per unit time, ω = dθ⁄dt, and it is identical for every particle.

What changes with distance is linear speed: v = ωr, so a particle twice as far from the axis moves twice as fast.

Why the others are wrong

• (a)
  Proportional to its distance from the axis — That describes linear speed, v = ωr, not angular velocity. A particle farther from the axis moves faster, but it sweeps the same angle in the same time.
• (c)
  Inversely proportional to its distance from the axis — Angular velocity does not fall off as 1⁄r. It is the same at every distance, while linear speed (ωr) and centripetal acceleration (ω²r) both grow with r.
• (d)
  Always zero — True only if the body is not rotating at all. A particle exactly on the axis has zero linear speed, but the body still turns with angular velocity ω.

For a rigid body rotating about a fixed axis, every particle moves in a circle centred on the axis, and all of them sweep equal angles in equal times.

So angular velocity ω belongs to the whole body, while each particle's linear speed depends on its distance r from the axis: v = ωr.

Acceleration splits the same way. Centripetal acceleration, ω²r, also grows with distance from the axis.

Key facts

• Angular velocity ω = dθ⁄dt, in rad/s, is the same for every particle of a rigid body rotating about a fixed axis.
• Linear speed v = ωr is proportional to the particle's distance r from the axis.
• Centripetal acceleration of a particle is ω²r, which equals v²⁄r.
• Earth's rotation shows it: angular speed is the same everywhere on the surface, but linear speed is greatest at the equator and near zero at the poles.

Common traps

• Mixing up angular and linear speed — the outer particle moves faster but turns through the same angle.
• Reading "rigid" as "stationary" and choosing "always zero".

No directly related past PYQ was found.