Angular Velocity — Interactive Module

Screen 01 · Concept of Angular Displacement

Concept Panel

Angular Displacement (θ)

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

Units

radian (SI) degree

Key Relation

θ (rad) = arc length / radius · 180° = π rad

Interactive Canvas

↻ drag radius to rotate

θ = 0.00 rad

θ = 0.0°

Screen 02 · Defining Angular Velocity

Definitions

Average Angular Velocity

ω̄ = Δθ / Δt

Total angular displacement divided by total time interval.

Instantaneous Angular Velocity

ω = dθ / dt

Limit of average angular velocity as Δt → 0.

Units

rad/s rev/min deg/s

Interactive Canvas

Δθ = 1.57 rad

Δt = 0.50 s

ω = 3.14 rad/s

Interactive

Slider: change Δθ and Δt
Observe ω update in real time

Controls

Δθ (radians)

current: 1.57 rad

Δt (seconds)

current: 0.50 s

Screen 03 · 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 ω⃗.

↺ Anticlockwise

↻ Clockwise

Sign Convention

⊙ out of screen→ anticlockwise ω

⊗ into screen→ clockwise ω

Interactive Canvas

ω⃗ (out of screen)

View

Screen 04 · Relation Between Linear & Angular Velocity

Key Relation

Linear ↔ Angular

v = r ω

v = linear (tangential) speed

r = radius (distance from axis)

ω = angular speed (rad/s)

Insight

All points on a rigid rotating body share the same ω, but points farther from the axis have larger linear speed v.

Interactive Canvas

Same ω · Different r

Interactive

Drag points to change r₁, r₂
Slider sets shared ω
v₁ and v₂ compared live

Controls

ω (rad/s)

ω = 4.0 rad/s

r₁ (m)

r₁ = 0.20 m

r₂ (m)

r₂ = 0.45 m

Comparison

v₁ (r₁ = 0.20 m) = 0.80 m/s

v₂ (r₂ = 0.45 m) = 1.80 m/s

v₁/v₂ = r₁/r₂ = 0.44

Screen 05 · Multiple Representations

MOTION ANIMATION

Challenge A · Motion → Graph

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

A

B

C

D

Challenge B · Motion → Graph

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

A

B

C

D

Screen 06 · JEE Main Practice — 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 · JEE Main Practice — Angular Velocity

Question

Q2.

The angular position of a rotating disc is given by θ(t) = 2t³ − 3t² + 4 (in radians, t in seconds). The angular velocity of the disc at t = 2 s is:

θ(t) = 2t³ − 3t² + 4

💡 Hint: ω = dθ/dt