Angular Velocity Β· Interactive Concept Module Screen 1 of 8
Screen 01 Β· Concept of Angular Displacement
Concept Panel

Angular Displacement (ΞΈ)

Angle swept by the radius vector of a particle in circular motion, measured from a reference line.

Units
radian (SI) degree
Key Relation
ΞΈ (rad) = arc length / radius  Β·  180Β° = Ο€ rad
Interactive Canvas
↻ drag radius to rotate
ΞΈ = 0.00 rad
ΞΈ = 0.0Β°
Screen 02 Β· Defining Angular Velocity
Definitions

Average Angular Velocity

Ο‰Μ„ = Δθ / Ξ”t

Total angular displacement divided by total time interval.

Instantaneous Angular Velocity

Ο‰ = dΞΈ / dt

Limit of average angular velocity as Ξ”t β†’ 0.

Units
rad/s rev/min deg/s
Interactive Canvas
Δθ = 1.57 rad
Ξ”t = 0.50 s
Ο‰ = 3.14 rad/s
Interactive
  • Slider: change Δθ and Ξ”t
  • Observe Ο‰ update in real time
Controls
current: 1.57 rad
current: 0.50 s
Screen 03 Β· Direction of Angular Velocity
Concept Panel

Right-Hand Thumb Rule

Curl the fingers of your right hand in the direction of rotation. The thumb points along the direction of Ο‰βƒ—.

β†Ί Anticlockwise
↻ Clockwise
Sign Convention
βŠ™ out of screenβ†’ anticlockwise Ο‰
βŠ— into screenβ†’ clockwise Ο‰
Interactive Canvas
Ο‰βƒ— (out of screen)
View
Screen 04 Β· Relation Between Linear & Angular Velocity
Key Relation

Linear ↔ Angular

v = r Ο‰

v = linear (tangential) speed

r = radius (distance from axis)

Ο‰ = angular speed (rad/s)

Insight
All points on a rigid rotating body share the same Ο‰, but points farther from the axis have larger linear speed v.
Interactive Canvas
Same Ο‰ Β· Different r
Interactive
  • Drag points to change r₁, rβ‚‚
  • Slider sets shared Ο‰
  • v₁ and vβ‚‚ compared live
Controls
Ο‰ = 4.0 rad/s
r₁ = 0.20 m
rβ‚‚ = 0.45 m
Comparison
v₁ (r₁ = 0.20 m) = 0.80 m/s
vβ‚‚ (rβ‚‚ = 0.45 m) = 1.80 m/s
v₁/vβ‚‚ = r₁/rβ‚‚ = 0.44
Screen 05 Β· Multiple Representations
MOTION ANIMATION
Challenge A Β· Motion β†’ Graph

Watch the rotation. Pick the matching θ–t graph.

A
B
C
D
Challenge B Β· Motion β†’ Graph

Watch the rotation above. Pick the matching ω–t graph.

A
B
C
D
Screen 06 Β· JEE Main Practice β€” Angular Velocity
Question

Q1.

A particle moves in a circle of radius 0.5 m with a constant angular speed. It completes 30 revolutions per minute. The linear speed of the particle is approximately:

Figure: particle on a horizontal circle, radius r = 0.5 m
Screen 07 Β· JEE Main Practice β€” Angular Velocity
Question

Q2.

The angular position of a rotating disc is given by ΞΈ(t) = 2tΒ³ βˆ’ 3tΒ² + 4 (in radians, t in seconds). The angular velocity of the disc at t = 2 s is:

ΞΈ(t) = 2tΒ³ βˆ’ 3tΒ² + 4
πŸ’‘ Hint: Ο‰ = dΞΈ/dt