Angular Displacement

Angular displacement describes how far an object has rotated and in which direction it has rotated.

Instead of measuring motion along a straight line in metres, rotational motion can be described using an angle.

Angular displacement is commonly represented by the Greek letter:

θ (theta)

It can be measured in:

• degrees (°)
• radians (rad)
• revolutions

For example, if a wheel turns through one-quarter of a complete rotation, its angular displacement is:

90° = π/2 rad

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Angular Position

Before angular displacement can be understood, it is useful to define angular position.

Angular position describes the orientation of an object relative to a chosen reference line.

For example, imagine a rotating wheel with a mark on its edge.

If the mark begins on the positive horizontal axis, we could define:

θ = 0°

After one-quarter turn counterclockwise:

θ = 90°

After half a turn:

θ = 180°

After three-quarters of a turn:

θ = 270°

Angular position tells us where the rotating object is relative to the chosen reference direction.

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Angular Displacement

Angular displacement is the change in angular position.

Therefore:

Δθ = θ_f − θ_i

where:

• Δθ = angular displacement
• θ_i = initial angular position
• θ_f = final angular position

This is similar to linear displacement:

Δx = x_f − x_i

Angular displacement describes a change in orientation rather than a change in linear position.

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Worked Example: Change in Angular Position

An object begins at:

θ_i = 30°

and rotates to:

θ_f = 110°

Then:

Δθ = θ_f − θ_i

Δθ = 110° − 30°

Δθ = 80°

The angular displacement is 80° in the positive rotational direction.

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Direction of Angular Displacement

Angular displacement includes direction.

A common sign convention is:

counterclockwise = positive

clockwise = negative

Therefore:

a 60° counterclockwise rotation can be written as:

Δθ = +60°

A 60° clockwise rotation can be written as:

Δθ = −60°

The sign convention should always be stated or understood before solving a problem.

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Worked Example: Clockwise Rotation

A wheel rotates clockwise through 135°.

Using the usual convention:

clockwise = negative

Therefore:

Δθ = −135°

The negative sign describes direction.

It does not mean the wheel rotated through a "negative amount" of angle.

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Degrees

A complete revolution is divided into:

360°

Therefore:

Degrees are familiar and convenient for describing angles, but physics often uses another unit:

radians.

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Radians

A radian is the SI unit used for angular displacement.

One complete revolution is:

2π rad

Since one complete revolution is also 360°:

360° = 2π rad

Therefore:

180° = π rad

and:

90° = π/2 rad

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What Is One Radian?

Consider a circle with radius r.

Imagine an arc along the circumference whose length s is exactly equal to the radius.

When:

s = r

the angle at the centre is:

1 rad

This gives the important relationship:

θ = s/r

when θ is measured in radians.

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Angular Displacement and Arc Length

For circular motion:

s = rθ

where:

• s = arc length or linear distance travelled along the circular path
• r = radius
• θ = angular displacement in radians

This equation creates a direct connection between angular and linear motion.

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Why Radians Are Useful

Radians make relationships such as:

s = rθ

very simple.

If degrees were used directly, an additional conversion factor would be required.

For this reason, rotational physics calculations generally use radians.

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Converting Degrees to Radians

Since:

180° = π rad

we can convert degrees to radians using:

radians = degrees × π/180

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Example: 60° to Radians

θ = 60° × π/180

θ = π/3

Therefore:

60° = π/3 rad

or approximately:

1.05 rad

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Example: 225° to Radians

θ = 225° × π/180

Simplify:

θ = 5π/4

Therefore:

225° = 5π/4 rad

≈ 3.93 rad

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Converting Radians to Degrees

Use:

degrees = radians × 180/π

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Example: π/6 rad

θ = π/6 × 180/π

θ = 30°

Therefore:

π/6 rad = 30°

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Example: 2.5 rad

θ = 2.5 × 180/π

θ ≈ 143.2°

Therefore:

2.5 rad ≈ 143°

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Common Degree-Radian Values

These common values are useful in rotational motion.

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Revolutions and Angular Displacement

Angular displacement can also be described using revolutions.

One revolution is:

1 rev = 360° = 2π rad

Therefore:

number of revolutions = θ/(2π)

when θ is measured in radians.

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Worked Example: Revolutions to Radians

A wheel completes:

3 revolutions

Then:

θ = 3(2π)

θ = 6π rad

In degrees:

θ = 3(360°)

θ = 1080°

Angular displacement can therefore be greater than 360°.

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Angular Displacement Greater Than One Revolution

An angular position is often represented between 0° and 360°.

Angular displacement, however, can include multiple rotations.

Suppose a wheel begins at 0° and makes:

2.5 counterclockwise revolutions

Its final orientation corresponds to 180°.

But its total angular displacement is:

2.5 × 360°

= 900°

or:

2.5 × 2π

= 5π rad

This distinction is important.

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Final Orientation Is Not Always Angular Displacement

Suppose a wheel begins with a mark pointing upward.

It completes one full revolution and returns to exactly the same orientation.

Its initial and final orientations are identical.

However, if we track the rotation continuously, the wheel has rotated through:

360° = 2π rad

This illustrates an important distinction between:

• orientation
• angular displacement accumulated during a specified rotation

The path and sign convention must be clear when interpreting rotational motion.

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Angular Displacement Versus Linear Displacement

Linear displacement describes the change in an object's position:

Δx = x_f − x_i

Angular displacement describes the change in rotational position:

Δθ = θ_f − θ_i

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Different Points on a Rotating Object

Consider a rigid rotating wheel.

Every point on the wheel rotates through the same angular displacement.

However, different points may travel different linear distances.

Suppose two points are located at:

Point A:

r = 0.10 m

Point B:

r = 0.30 m

Both rotate through:

θ = 2 rad

For Point A:

s = rθ

s = (0.10)(2)

s = 0.20 m

For Point B:

s = (0.30)(2)

s = 0.60 m

Both points have the same angular displacement but different linear displacements along their circular paths.

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Why Outer Points Travel Farther

For the same angular displacement:

s = rθ

Therefore:

s ∝ r

A point twice as far from the axis travels twice the arc length.

A point three times as far from the axis travels three times the arc length.

This explains why the outer edge of a rotating wheel travels farther than a point near the centre during the same rotation.

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Worked Example: Arc Length

A wheel has radius:

r = 0.40 m

It rotates through:

θ = 2.5 rad

Calculate the distance travelled by a point on its edge.

Use:

s = rθ

s = (0.40)(2.5)

s = 1.0 m

The point travels 1.0 m along the circular path.

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Worked Example: Finding Angular Displacement

A point on the edge of a wheel travels:

s = 1.8 m

The wheel has radius:

r = 0.30 m

Use:

θ = s/r

θ = 1.8/0.30

θ = 6.0 rad

In revolutions:

number of revolutions = 6.0/(2π)

≈ 0.955 revolutions

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Angular Position on a Circle

Angular position can be used like a coordinate for rotational motion.

Suppose counterclockwise is positive.

A rotating object might have:

at t = 0 s:

θ = 0 rad

at t = 1 s:

θ = 0.5 rad

at t = 2 s:

θ = 1.2 rad

at t = 3 s:

θ = 2.0 rad

The increasing angular position shows that the object is rotating counterclockwise.

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Interpreting Angular Position Data

Consider:

The angular position increases by:

1 rad each second

This indicates uniform rotational motion.

The object has accumulated:

4 rad

of angular displacement after 4 seconds.

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Angular Position-Time Graphs

Rotational motion can be represented using a graph of:

angular position θ

against:

time t

This is similar to a linear position-time graph.

If θ increases steadily:

the object rotates in the positive direction.

If θ decreases:

the object rotates in the negative direction.

If θ remains constant:

the object is not rotating.

A steeper graph indicates a greater rate of change of angular position.

This leads naturally to the idea of angular velocity.

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Calculating Angular Displacement from Angular Position

Suppose:

θ_i = 1.5 rad

θ_f = 5.0 rad

Then:

Δθ = θ_f − θ_i

Δθ = 5.0 − 1.5

Δθ = +3.5 rad

The positive sign indicates rotation in the positive direction.

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Negative Angular Displacement

Suppose:

θ_i = 2.0 rad

θ_f = −1.0 rad

Then:

Δθ = −1.0 − 2.0

Δθ = −3.0 rad

The negative sign indicates rotation in the negative direction.

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Worked Example: Degrees

A rotating platform begins at:

θ_i = 40°

and rotates counterclockwise to:

θ_f = 250°

Then:

Δθ = 250° − 40°

Δθ = +210°

Convert to radians:

Δθ = 210 × π/180

Δθ = 7π/6 rad

≈ 3.67 rad

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Rotational Direction Matters

Suppose a wheel starts at 20° and finishes at 80°.

One possible angular displacement is:

+60°

if it rotates counterclockwise directly from 20° to 80°.

But the wheel could instead rotate clockwise through:

−300°

and arrive at the same orientation.

Therefore, initial and final orientations alone may not always tell us the actual angular displacement.

We may also need to know:

the direction and path of rotation.

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Multiple Revolutions

Suppose a motor turns a shaft through:

12 complete revolutions counterclockwise

Then:

θ = 12(2π)

θ = 24π rad

≈ 75.4 rad

Angular displacement is not restricted to values between 0 and 2π.

This is important when studying:

• motors
• turbines
• wheels
• gears
• engines

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Angular Displacement of a Clock Hand

The hands of a clock provide a familiar example of rotational motion.

The minute hand completes:

360°

every:

60 minutes

Therefore, in 15 minutes it rotates:

15/60 × 360°

= 90°

Because a clock hand moves clockwise, using the standard sign convention:

Δθ = −90°

or:

Δθ = −π/2 rad

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Worked Example: Minute Hand

How far does the minute hand rotate in 20 minutes?

Fraction of revolution:

20/60 = 1/3

Therefore:

θ = 1/3(360°)

= 120°

In radians:

θ = 1/3(2π)

= 2π/3 rad

Since the rotation is clockwise:

Δθ = −2π/3 rad

if counterclockwise is defined as positive.

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Wheels and Vehicles

Angular displacement can be connected to the distance travelled by a rolling wheel.

If a wheel rolls without slipping:

s = rθ

where s is the distance travelled by the vehicle.

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Worked Example: Bicycle Wheel

A bicycle wheel has radius:

0.35 m

It rotates through:

20 rad

Then:

s = rθ

s = (0.35)(20)

s = 7.0 m

The bicycle travels 7.0 m if the wheel rolls without slipping.

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Finding Wheel Rotation from Distance

A car wheel has radius:

0.30 m

The car travels:

60 m

Assuming no slipping:

θ = s/r

θ = 60/0.30

θ = 200 rad

Number of revolutions:

N = θ/(2π)

N = 200/(2π)

≈ 31.8 revolutions

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Gears and Angular Displacement

Connected gears provide another useful application.

When two gears mesh, the edges move through the same linear distance at their point of contact.

Therefore:

r₁θ₁ = r₂θ₂

The larger gear rotates through a smaller angle while the smaller gear rotates through a larger angle.

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Worked Gear Example

Gear A has radius:

r_A = 5 cm

Gear B has radius:

r_B = 15 cm

Gear A rotates through:

θ_A = 6 rad

Using:

r_A θ_A = r_B θ_B

(5)(6) = (15)θ_B

θ_B = 2 rad

The gears rotate in opposite directions.

So if Gear A rotates:

+6 rad

then Gear B rotates:

−2 rad

using the usual sign convention.

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Rotating Machinery

Angular displacement is important in:

• electric motors
• generators
• turbines
• fans
• engines
• robotics
• manufacturing equipment

Sensors called rotary encoders can measure angular position.

A control system can then determine how far a shaft has rotated.

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Robotics

Robotic joints often rotate rather than move in straight lines.

A robotic arm might be instructed to rotate a joint from:

θ_i = 25°

to:

θ_f = 70°

Then:

Δθ = 70° − 25°

Δθ = 45°

Precise angular positioning allows robots to control the location of their arms and tools.

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Angular and Linear Motion Together

Rotational motion has quantities that correspond closely to linear motion.

Understanding angular displacement is therefore the foundation for studying:

• angular velocity
• angular acceleration
• rotational dynamics

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Radians in Rotational Physics

Although angles can be measured in degrees, equations connecting angular and linear quantities normally require radians.

For example:

s = rθ

works directly when θ is measured in radians.

Later relationships such as:

v = rω

and:

a_t = rα

also use angular quantities measured in radians.

This is one reason radians are the standard unit in rotational physics.

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A Problem-Solving Strategy

For angular displacement problems, first identify:

initial angular position θ_i

and:

final angular position θ_f

Then use:

Δθ = θ_f − θ_i

Check the sign:

• positive → counterclockwise
• negative → clockwise

If necessary, convert units:

180° = π rad

For circular distance problems, use:

s = rθ

with θ in radians.

For complete revolutions:

θ = 2πN

where N is the number of revolutions.

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Common Mistakes

Confusing angular position with angular displacement

Angular position describes orientation. Angular displacement describes the change in angular position.

Ignoring direction

Angular displacement can be positive or negative.

Assuming clockwise is positive

The usual convention is counterclockwise positive, although a different convention can be chosen if clearly stated.

Using degrees in s = rθ

The equation requires θ in radians.

Thinking one revolution equals π radians

One complete revolution is:

2π rad

Thinking angular displacement must be less than 360°

Objects can rotate through many complete revolutions.

Confusing angular and linear displacement

Points on the same rigid rotating object have the same angular displacement but can travel different linear distances.

Forgetting the radius

Linear distance around a circle depends on both angular displacement and radius.

Assuming the final orientation tells the complete rotational history

The same final orientation can result from many different angular displacements.

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Key Terms

Angular position (θ) — orientation of an object relative to a chosen angular reference.

Angular displacement (Δθ) — change in angular position.

Degree (°) — angular unit in which one complete revolution equals 360°.

Radian (rad) — SI unit of angular measurement defined using arc length and radius.

Revolution (rev) — one complete rotation, equal to 360° or 2π rad.

Reference line — chosen direction from which angular position is measured.

Positive rotation — usually counterclockwise rotation.

Negative rotation — usually clockwise rotation.

Arc length (s) — distance measured along a circular path.

Radius (r) — distance from the axis of rotation to a point on the rotating object.

Angular coordinate — angular value used to describe rotational position.

Rotational motion — motion of an object around an axis.

Rigid body — object whose points maintain fixed distances from one another during rotation.

Rotary encoder — sensor used to measure angular position or rotation.

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Key Takeaways

• Angular displacement describes the amount and direction of rotation.
• Angular position is represented by θ.
• Angular displacement is calculated using Δθ = θ_f − θ_i.
• Counterclockwise rotation is usually positive.
• Clockwise rotation is usually negative.
• One complete revolution is 360° = 2π rad.
• Half a revolution is 180° = π rad.
• A quarter revolution is 90° = π/2 rad.
• Radians are the standard angular unit used in rotational physics.
• Degrees can be converted using radians = degrees × π/180.
• Radians can be converted using degrees = radians × 180/π.
• Angular displacement can be greater than 360° or 2π rad.
• Points on the same rigid rotating object experience the same angular displacement.
• Points farther from the axis travel greater linear distances.
• Angular and linear displacement are connected by s = rθ.
• The equation s = rθ requires θ in radians.
• Rolling wheels connect rotational motion to linear travel.
• Angular position-time data can be used to describe rotational motion.
• The same final orientation can sometimes result from different angular displacements.
• Angular displacement provides the foundation for understanding angular velocity and angular acceleration.

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Check Your Understanding

1. Define angular displacement and explain how it differs from angular position.
2. State the usual sign convention for clockwise and counterclockwise rotation.
3. Convert 120° to radians.
4. Convert 5π/6 rad to degrees.
5. A wheel rotates counterclockwise through 2.5 revolutions. Determine its angular displacement in degrees and radians.
6. An object moves from angular position θ_i = 0.8 rad to θ_f = 4.2 rad. Calculate its angular displacement.
7. A wheel begins at 150° and rotates clockwise through 220°. State its angular displacement using the usual sign convention.
8. A point on a rotating disk is 0.25 m from the axis. The disk rotates through 4 rad. Calculate the distance travelled by the point along its circular path.
9. Two points are located 0.10 m and 0.40 m from the centre of a rotating disk. The disk rotates through 3 rad. Compare their angular displacements and calculate the distance travelled by each point.
10. A car wheel has radius 0.32 m and rotates through 150 rad without slipping. Calculate the distance travelled by the car.
11. A wheel of radius 0.20 m rolls 25 m without slipping. Calculate its angular displacement in radians and the number of complete revolutions.
12. The minute hand of a clock moves from 12:00 to 12:35. Calculate its angular displacement in degrees and radians using counterclockwise as positive.
13. Gear A has radius 4 cm and rotates through 10 rad. It drives Gear B with radius 20 cm. Calculate the magnitude of Gear B's angular displacement and state its direction relative to Gear A.
14. Explain why a point near the edge of a rotating wheel travels farther than a point near the centre even though both have the same angular displacement.
15. A wheel begins with a mark at 30° and finishes with the mark at 90°. Explain why this information alone may not be enough to determine the wheel's actual angular displacement.