The angle at the centre of a circle subtended by an arc equal in length to the radius. It is the SI unit of angle; 1 revolution = 2π rad = 360°, so 1 rad ≈ 57.3°.

How is the direction of angular displacement found?

By the right-hand rule: curl the fingers of the right hand in the sense of rotation; the thumb points along the axis in the direction of the angular displacement. Counterclockwise rotation is taken as positive.

How are linear and angular quantities related?

S = rθ, v = rω and a = rα, with θ in radians. Points farther from the axis move faster, but all points of a rigid body share the same ω and α.

What are the equations of angular motion?

ω_f = ω_i + αt, 2αθ = ω_f² − ω_i², and θ = ω_i t + ½αt². They hold when the axis of rotation is fixed.

Which board sets my paper if I study in Gujrat?

Gujrat district is listed under the Board of Intermediate and Secondary Education, Gujranwala. PECTAA writes the books. Your board sets the paper and the dates, so follow its notices.

Angular Displacement, Velocity, Acceleration and Equations of Angular Motion: PECTAA Class 11 Physics, Punjab Board

Switch off a ceiling fan and it keeps turning, slower and slower, for many seconds. How many turns does it make before it stops? The equations you already know for straight-line motion answer this once you swap distance, velocity and acceleration for their angular partners.

Chapter 3 of the PECTAA Physics 11 book is Circular and Rotational Motion. After Chapter 2 (Force and Motion), which ended with collisions in two dimensions and rocket propulsion, section 3.1 covers angular measurements and the equations of angular motion. This note is for FSc Part 1 (1st year, Class 11) students in Lahore, Faisalabad, Sargodha, Rawalpindi, Multan, Gujranwala and the other Punjab boards. Gujrat district comes under the Gujranwala board. PECTAA writes the books. Your own board sets your paper.

What to know

Radian (section 3.1). If an arc AB equals the radius r, the angle it makes at the centre is one radian (rad), the SI unit of angle. For an arc of length S,

S = rθ (Eq. 3.1), θ in radians.

One revolution covers S = 2πr, so 1 rev = 2π rad = 360°, and 1 rad ≈ 57.3°.

Angular displacement. Picture a particle P on a light rigid rod pivoted at O. If OP turns from angle θ to θ + Δθ in time Δt, Δθ is the angular displacement. It is positive for counterclockwise rotation. For very small values it is a vector along the axis of rotation, with direction given by the right-hand rule.

Angular velocity. Average: ω_av = Δθ/Δt (Eq. 3.2). Instantaneous: the limit as Δt → 0 (Eq. 3.3). The SI unit is rad s⁻¹; revolutions per minute (rpm) are also used.

Angular acceleration. The rate of change of angular velocity. Average: α_av = (ω₂ − ω₁)/(t₂ − t₁) (Eq. 3.4); instantaneous: the limit as Δt → 0 (Eq. 3.5). Unit: rad s⁻². A fan speeding up after being switched on has angular acceleration.

Rigid bodies. A rigid body is described by a reference line OP drawn from the axis to any point P. Every reference line in the body turns with the same ω and α.

Linear and angular velocity. A point P at distance r from the axis moves a distance ΔS = rΔθ. Dividing by Δt and taking the limit:

v = rω (Eq. 3.7).

v points along the tangent, so it is called tangential velocity. Similarly the tangential acceleration is

a = rα (Eq. 3.8).

So on a turning body, points at different distances have different speeds, but all share the same θ, ω and α.

Equations of angular motion. Replace S, v and a in the linear equations by θ, ω and α:

Linear	Angular
v_f = v_i + at	ω_f = ω_i + αt (3.9)
2aS = v_f² − v_i²	2αθ = ω_f² − ω_i² (3.10)
S = v_i t + ½at²	θ = ω_i t + ½αt² (3.11)

They hold only when the axis of rotation is fixed, so all angular vectors point the same way and can be treated as scalars.

Key points

Use radians in S = rθ, v = rω and a = rα.
1 rev = 2π rad = 360°.
Same ω for the whole rigid body; v grows with r.
Angular equations need a fixed axis.

Worked examples

Example 1 (book Example 3.1). A fan rotating at 3 rev s⁻¹ is switched off and comes to rest in 18.0 s. Find the uniform deceleration and the number of revolutions.

ω_i = 3.0 rev s⁻¹, ω_f = 0, t = 18.0 s.
α = (ω_f − ω_i)/t = (0 − 3.0)/18.0 ≈ −0.167 rev s⁻².
θ = ω_i t + ½αt² = 3.0 × 18.0 + ½ × (−0.167) × (18.0)² = 54 − 27 = 27 rev.

Example 2 (unit change). Express 3 rev s⁻¹ in rad s⁻¹.

1 rev = 2π rad.
ω = 3 × 2π = 6π ≈ 18.85 rad s⁻¹.

Example 3 (v = rω). A wheel of radius 0.5 m turns at 4 rad s⁻¹. Find the speed of a point on its rim.

v = rω = 0.5 × 4 = 2 m s⁻¹.

Common mistakes in the exam

Using degrees in S = rθ. θ must be in radians.
Forgetting the ½ in θ = ω_i t + ½αt².
Giving different ω values to points on one rigid body.
Ignoring the sign of α for slowing down.
Mixing rev and rad in one equation without converting.

Exam technique

List known values (ω_i, ω_f, α, t, θ) and choose the equation that fits, as the book does.
Keep units consistent: all rev or all rad.
State the right-hand rule in words when asked for direction.
Check your board's notice for the sections in your paper.

Practice questions

Convert 180° to radians. Answer: π rad.
An arc of 6 m subtends 2 rad. Find the radius. Answer: r = S/θ = 3 m.
A wheel speeds up from 2 to 10 rad s⁻¹ in 4 s. Find α. Answer: (10 − 2)/4 = 2 rad s⁻².
With the same values, find θ. Answer: 2 × 4 + ½ × 2 × 16 = 24 rad.
Two points are 0.1 m and 0.3 m from the axis of a disc. Compare their speeds. Answer: The outer point moves 3 times faster; ω is the same.

Ready to practise this? Continue in 1st Year Physics.