Rotational Motion Equations

Rotational motion can be analyzed using a set of rotational kinematic equations. These equations describe how angular position, angular velocity, and angular acceleration change with time.

They closely resemble the equations used for linear motion.

The key difference is that linear quantities are replaced by their rotational equivalents:

• displacement Δx → angular displacement Δθ
• velocity v → angular velocity ω
• acceleration a → angular acceleration α

When angular acceleration is constant, the same mathematical patterns used in linear kinematics can be applied to rotation.

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The Main Rotational Variables

Before choosing an equation, identify the quantities involved.

Angular displacement is:

Δθ = θ_f − θ_i

These variables form the basis of rotational kinematics.

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Sign Conventions

Direction must be included when solving rotational-motion problems.

A common convention is:

counterclockwise = positive

clockwise = negative

Therefore:

• +ω → counterclockwise angular velocity
• −ω → clockwise angular velocity
• +α → angular velocity changing in the positive direction
• −α → angular velocity changing in the negative direction

The sign of α does not automatically tell you whether the object is speeding up or slowing down.

The signs of both ω and α must be considered.

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The Constant-Angular-Acceleration Equations

When angular acceleration is constant, four main equations are particularly useful.

Angular velocity and time

ω_f = ω_i + αt

Angular displacement and time

Δθ = ω_i t + 1/2 αt²

Angular velocity and displacement

ω_f² = ω_i² + 2αΔθ

Average angular velocity

Δθ = [(ω_i + ω_f)/2]t

These equations apply when:

α = constant

They should not be applied directly when angular acceleration varies significantly with time.

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Comparing Linear and Rotational Equations

The rotational equations are almost identical to the familiar linear kinematic equations.

This correspondence is extremely useful.

If you understand constant-acceleration linear motion, much of rotational kinematics follows the same mathematical structure.

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Why the Equations Are Similar

Consider linear motion:

a = Δv/Δt

For rotational motion:

α = Δω/Δt

Both describe how a velocity changes with time.

Similarly:

v = Δx/Δt

corresponds to:

ω = Δθ/Δt

Rotation therefore has its own versions of position, velocity and acceleration.

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Choosing the Correct Equation

A good strategy is to list the five major quantities:

ω_i, ω_f, α, Δθ, t

Then identify:

• what you know
• what you need to find
• which quantity is not involved

Choose an equation that contains the known quantities and the unknown you want.

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Equation Selection Table

The best equation is usually the one containing only one unknown.

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Angular Velocity and Time

The first rotational kinematic equation is:

ω_f = ω_i + αt

It describes how angular velocity changes when angular acceleration is constant.

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Worked Example: Motor Speeding Up

A motor initially rotates at:

ω_i = 5 rad/s

It accelerates at:

α = 3 rad/s²

for:

t = 4 s

Calculate its final angular velocity.

Use:

ω_f = ω_i + αt

ω_f = 5 + (3)(4)

ω_f = 17 rad/s

Therefore:

ω_f = 17 rad/s

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Worked Example: Slowing Wheel

A wheel rotates at:

ω_i = 30 rad/s

and experiences:

α = −5 rad/s²

for:

4 s

Then:

ω_f = 30 + (−5)(4)

ω_f = 10 rad/s

Therefore:

ω_f = 10 rad/s

The wheel is still rotating in the positive direction, but it has slowed.

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Finding Angular Acceleration

The same equation can be rearranged:

α = (ω_f − ω_i)/t

Suppose a fan speeds up from:

10 rad/s

to:

40 rad/s

in:

6 s

Then:

α = (40 − 10)/6

α = 5 rad/s²

Therefore:

α = 5 rad/s²

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Finding Time

The equation can also be rearranged:

t = (ω_f − ω_i)/α

Suppose:

ω_i = 8 rad/s

ω_f = 32 rad/s

α = 6 rad/s²

Then:

t = (32 − 8)/6

t = 4 s

Therefore:

t = 4 s

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Angular Displacement and Time

When initial angular velocity, angular acceleration and time are known:

Δθ = ω_i t + 1/2 αt²

This equation includes two contributions to angular displacement.

ω_i t

represents the displacement that would occur because of the initial angular velocity.

1/2 αt²

represents the additional effect of angular acceleration.

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Worked Example: Accelerating Wheel

A wheel initially rotates at:

ω_i = 4 rad/s

It accelerates at:

α = 2 rad/s²

for:

t = 5 s

Calculate its angular displacement.

Δθ = ω_i t + 1/2 αt²

Δθ = (4)(5) + 1/2(2)(5²)

Δθ = 20 + 25

Δθ = 45 rad

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Starting from Rest

If an object begins from rest:

ω_i = 0

Then:

ω_f = αt

and:

Δθ = 1/2 αt²

These simplified equations are useful for motors, wheels and machinery that begin stationary.

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Worked Example: Turntable Starting from Rest

A turntable starts from rest and accelerates at:

α = 2.5 rad/s²

for:

6 s

Final angular velocity:

ω_f = αt

ω_f = (2.5)(6)

ω_f = 15 rad/s

Angular displacement:

Δθ = 1/2 αt²

Δθ = 1/2(2.5)(6²)

Δθ = 45 rad

Therefore:

Δθ = 45 rad

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Converting Angular Displacement to Revolutions

Angular displacement is usually calculated in radians.

One complete revolution is:

2π rad

Therefore:

N = Δθ/(2π)

where N is the number of revolutions.

For the previous example:

N = 45/(2π)

N ≈ 7.16 revolutions

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Angular Velocity and Displacement

If time is not known, use:

ω_f² = ω_i² + 2αΔθ

This equation is particularly useful when a question asks:

• how far an object rotates before stopping
• how much rotation occurs before reaching a certain angular velocity
• what angular acceleration is required over a particular angle

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Worked Example: Finding Angular Displacement

A wheel speeds up from:

ω_i = 6 rad/s

to:

ω_f = 18 rad/s

with constant angular acceleration:

α = 4 rad/s²

Use:

ω_f² = ω_i² + 2αΔθ

18² = 6² + 2(4)Δθ

324 = 36 + 8Δθ

288 = 8Δθ

Δθ = 36 rad

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Worked Example: Braking Distance in Rotation

A machine shaft rotates at:

ω_i = 40 rad/s

It is slowed with:

α = −8 rad/s²

How far does it rotate before stopping?

At rest:

ω_f = 0

Use:

ω_f² = ω_i² + 2αΔθ

0² = 40² + 2(−8)Δθ

0 = 1600 − 16Δθ

16Δθ = 1600

Δθ = 100 rad

Number of revolutions:

N = 100/(2π)

≈ 15.9 revolutions

The shaft completes almost 16 additional revolutions before stopping.

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Average Angular Velocity

For constant angular acceleration:

ω_avg = (ω_i + ω_f)/2

Angular displacement can therefore be calculated using:

Δθ = [(ω_i + ω_f)/2]t

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Worked Example

A fan increases its angular velocity uniformly from:

8 rad/s

to:

24 rad/s

during:

5 s

Average angular velocity:

ω_avg = (8 + 24)/2

ω_avg = 16 rad/s

Then:

Δθ = ω_avg t

Δθ = (16)(5)

Δθ = 80 rad

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Why the Average Equation Works

If angular acceleration is constant, angular velocity changes uniformly.

For example:

8 → 12 → 16 → 20 → 24 rad/s

The average of the initial and final angular velocities represents the average angular velocity over the interval.

Therefore:

ω_avg = (ω_i + ω_f)/2

This relationship does not generally apply if angular acceleration varies unpredictably.

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Rotational Motion Graphs

Rotational kinematic equations can also be understood graphically.

The three most useful graphs are:

• angular position-time
• angular velocity-time
• angular acceleration-time

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Angular Position-Time Graph

On a θ-t graph:

gradient = angular velocity

Therefore:

ω = gradient of θ-t graph

A straight line means constant angular velocity.

A curved line means angular velocity is changing.

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Angular Velocity-Time Graph

On an ω-t graph:

gradient = angular acceleration

Therefore:

α = gradient of ω-t graph

The area under the graph represents:

angular displacement

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Worked Graph Example

Suppose angular velocity increases uniformly from:

5 rad/s

to:

25 rad/s

over:

4 s

Angular acceleration:

α = (25 − 5)/4

α = 5 rad/s²

Angular displacement is the area under the ω-t graph:

Δθ = (5 + 25)/2

Δθ = (15)(4)

Δθ = 60 rad

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Angular Acceleration-Time Graph

On an α-t graph:

area = change in angular velocity

For constant angular acceleration:

Δω = αt

Suppose:

α = 3 rad/s²

for:

t = 6 s

Then:

Δω = (3)(6)

Δω = 18 rad/s

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Positive and Negative Motion

Correct use of signs is essential.

Suppose counterclockwise is positive.

A wheel has:

ω_i = +20 rad/s

α = −4 rad/s²

After 3 s:

ω_f = 20 + (−4)(3)

ω_f = +8 rad/s

The wheel is still rotating counterclockwise but slowing down.

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Reversing Direction

Continue the previous motion for:

7 s

Then:

ω_f = 20 + (−4)(7)

ω_f = −8 rad/s

The negative final angular velocity means the wheel has reversed direction and is now rotating clockwise.

It must have passed through:

ω = 0

during the motion.

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When Does It Reverse?

Use:

0 = 20 − 4t

4t = 20

t = 5 s

Therefore:

• 0–5 s → counterclockwise and slowing
• at 5 s → momentarily stationary
• after 5 s → clockwise and speeding up

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Angular Displacement During a Direction Reversal

For the same motion:

ω_i = 20 rad/s

α = −4 rad/s²

t = 7 s

Use:

Δθ = ω_i t + 1/2 αt²

Δθ = (20)(7) + 1/2(−4)(7²)

Δθ = 140 − 98

Δθ = +42 rad

The net angular displacement is +42 rad.

Notice that this is not the same as the total angular distance travelled because the wheel changed direction.

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Angular Displacement Versus Total Rotation

When an object reverses direction, distinguish between:

angular displacement

and:

total angle rotated

Angular displacement depends only on the net change in angular position.

Total angular distance adds all rotation regardless of direction.

This is similar to the distinction between displacement and distance in linear motion.

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Connecting Angular and Linear Motion

Rotational quantities can be connected to linear quantities.

Angular displacement:

s = rθ

Tangential speed:

v = rω

Tangential acceleration:

a_t = rα

These relationships allow rotational equations to describe real moving objects such as:

• wheels
• pulleys
• gears
• turbines
• motors

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Worked Example: Accelerating Bicycle Wheel

A bicycle wheel has radius:

r = 0.35 m

It starts from rest and has:

α = 6 rad/s²

for:

t = 3 s

Final Angular Velocity

ω_f = αt

ω_f = (6)(3)

ω_f = 18 rad/s

Final Linear Speed

v = rω

v = (0.35)(18)

v = 6.3 m/s

Tangential Acceleration

a_t = rα

a_t = (0.35)(6)

a_t = 2.1 m/s²

Angular Displacement

Δθ = 1/2 αt²

Δθ = 1/2(6)(3²)

Δθ = 27 rad

Distance Travelled

s = rθ

s = (0.35)(27)

s = 9.45 m

The rotational equations therefore describe the forward motion of the bicycle as well.

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Rolling Without Slipping

For a wheel rolling without slipping:

s = rθ

v = rω

a = rα

These relationships connect the rotation of the wheel with the translation of the vehicle.

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Worked Example: Car Accelerating

A car accelerates uniformly from rest at:

a = 2.5 m/s²

Its wheels have radius:

r = 0.40 m

Find the angular acceleration.

Use:

a = rα

α = a/r

α = 2.5/0.40

α = 6.25 rad/s²

After 8 seconds:

ω = αt

ω = (6.25)(8)

ω = 50 rad/s

The car's linear speed is:

v = rω

v = (0.40)(50)

v = 20 m/s

This agrees with linear kinematics:

v = at

v = (2.5)(8)

v = 20 m/s

Both approaches describe the same motion.

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Rotating Machinery

Rotational kinematic equations are frequently used to describe motors and machinery.

Consider an industrial motor that starts from rest and reaches:

180 rad/s

in:

12 s

Assuming constant angular acceleration:

α = (180 − 0)/12

α = 15 rad/s²

Angular displacement during startup:

Δθ = (0 + 180)/2

Δθ = 90(12)

Δθ = 1080 rad

Number of revolutions:

N = 1080/(2π)

≈ 172 revolutions

Even during a short startup, high-speed machinery may complete many revolutions.

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Braking a Rotating Machine

Suppose the same motor is rotating at:

180 rad/s

and must stop in:

15 s

Then:

α = (0 − 180)/15

α = −12 rad/s²

Angular displacement while stopping:

Δθ = (180 + 0)/2

Δθ = 90(15)

Δθ = 1350 rad

Number of revolutions:

N = 1350/(2π)

≈ 215 revolutions

A rapidly rotating machine can continue turning many times even while braking.

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Washing Machine Example

A washing-machine drum starts from rest and reaches:

ω_f = 100 rad/s

in:

20 s

Assuming constant angular acceleration:

α = (100 − 0)/20

α = 5 rad/s²

Angular displacement:

Δθ = (0 + 100)/2

Δθ = 50(20)

Δθ = 1000 rad

Number of revolutions:

N = 1000/(2π)

≈ 159 revolutions

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Converting rpm to Angular Velocity

Real machines often give rotational speed in:

revolutions per minute (rpm)

Convert using:

ω = rpm × 2π/60

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Worked Example: 1200 rpm

A motor rotates at:

1200 rpm

Then:

ω = 1200 × 2π/60

ω = 40π

ω ≈ 126 rad/s

If the motor reaches this speed from rest in 10 s:

α = 126/10

α ≈ 12.6 rad/s²

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Clock Hands

Rotational equations can also describe clock hands.

The minute hand completes:

2π rad

in:

3600 s

Its angular velocity magnitude is:

ω = 2π/3600

ω ≈ 1.75 × 10⁻³ rad/s

Because the motion is clockwise, if counterclockwise is positive:

ω ≈ −1.75 × 10⁻³ rad/s

Since the minute hand rotates at nearly constant angular velocity:

α ≈ 0

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Gears

For two meshing gears without slipping, the points where the gears touch have the same tangential speed.

Therefore:

r₁ω₁ = r₂ω₂

and for angular acceleration magnitudes:

r₁α₁ = r₂α₂

A smaller gear rotates faster and usually has a greater angular acceleration magnitude.

Meshing gears also rotate in opposite directions.

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Worked Gear Example

Gear A has radius:

r_A = 5 cm

Gear B has radius:

r_B = 20 cm

Gear A accelerates from rest at:

α_A = 12 rad/s²

Then:

r_A α_A = r_B |α_B|

(5)(12) = (20)|α_B|

|α_B| = 3 rad/s²

If Gear A's acceleration is positive:

α_B = −3 rad/s²

because Gear B accelerates rotationally in the opposite direction.

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Pulley Systems

Rotating pulleys provide another application.

If a rope does not slip on a pulley:

v = rω

and:

a = rα

Therefore, the motion of the rope can be related directly to the rotation of the pulley.

This becomes important when studying:

• elevators
• cranes
• belt drives
• mechanical lifting systems

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Selecting Equations Efficiently

Suppose a problem gives:

ω_i, ω_f and t

and asks for α.

Use:

α = (ω_f − ω_i)/t

Suppose it gives:

ω_i, α and t

and asks for Δθ.

Use:

Δθ = ω_i t + 1/2 αt²

Suppose it gives:

ω_i, ω_f and α

but no time.

Use:

ω_f² = ω_i² + 2αΔθ

The goal is not to memorize which equation belongs to which question.

Instead:

identify the variables first.

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A Useful Variable Table

For longer problems, make a table before calculating.

Fill in everything given by the problem.

Then select an equation containing:

• the required unknown
• the known quantities
• no unnecessary unknown

This greatly reduces equation-selection errors.

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Multi-Step Problems

Many real problems require more than one equation.

For example:

A wheel starts from rest, accelerates at 4 rad/s² for 5 s, and has radius 0.30 m.

Find its final linear speed.

First find angular velocity:

ω_f = ω_i + αt

ω_f = 0 + (4)(5)

ω_f = 20 rad/s

Then convert to linear speed:

v = rω

v = (0.30)(20)

v = 6.0 m/s

The problem combines rotational kinematics with the angular-linear relationship.

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Real-World Applications

Rotational kinematic equations are used in the analysis of:

• vehicle wheels
• electric motors
• turbines
• wind turbines
• generators
• robotic joints
• gears
• pulleys
• flywheels
• washing machines
• industrial equipment
• amusement rides

Engineers often need to determine how quickly rotating equipment can safely:

• start
• stop
• accelerate
• reverse direction

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Rotational Motion and Safety

A rotating machine cannot usually stop instantly.

Its angular velocity must decrease over time.

Rotational kinematics can determine:

• stopping time
• stopping angle
• number of revolutions before stopping

This information is important when designing:

• brakes
• emergency shutdown systems
• industrial machinery
• vehicles

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A Complete Worked Problem

A wheel initially rotates counterclockwise at:

ω_i = 10 rad/s

It accelerates uniformly at:

α = 4 rad/s²

for:

t = 6 s

The wheel has radius:

r = 0.25 m

Final Angular Velocity

ω_f = ω_i + αt

ω_f = 10 + (4)(6)

ω_f = 34 rad/s

Angular Displacement

Δθ = ω_i t + 1/2 αt²

Δθ = (10)(6) + 1/2(4)(6²)

Δθ = 60 + 72

Δθ = 132 rad

Number of Revolutions

N = 132/(2π)

N ≈ 21.0 revolutions

Final Tangential Speed

v = rω

v = (0.25)(34)

v = 8.5 m/s

Tangential Acceleration

a_t = rα

a_t = (0.25)(4)

a_t = 1.0 m/s²

One problem can therefore connect several rotational quantities.

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Checking Your Answer

After solving a rotational problem, ask:

Are the units correct?

• θ → rad
• ω → rad/s
• α → rad/s²
• t → s

Does the sign make sense?

If an object is slowing while rotating positively, α should be negative.

Is the magnitude reasonable?

A wheel should not suddenly gain thousands of rad/s unless the situation supports such a large acceleration.

Did I use radians where necessary?

Rotational equations normally use radians.

Did I assume constant α?

If the equation requires constant angular acceleration, make sure that assumption is justified.

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Common Mistakes

Mixing linear and rotational variables

Do not substitute v where ω is required or a where α is required.

Using degrees instead of radians

Rotational kinematic calculations should generally use radians.

Forgetting direction

Clockwise and counterclockwise quantities may require different signs.

Thinking negative α always means slowing down

Whether the object speeds up or slows down depends on both ω and α.

Using the wrong equation

List the known and unknown variables before choosing an equation.

Forgetting that 1 revolution = 2π rad

Do not treat revolutions and radians as interchangeable numbers.

Using rpm directly as rad/s

Convert before using rotational equations.

Assuming constant angular acceleration when it is not constant

These four main equations require constant α.

Confusing angular displacement with number of revolutions

Convert using:

N = Δθ/(2π)

Forgetting radius when converting to linear quantities

Use:

s = rθ

v = rω

a_t = rα

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Key Terms

Rotational kinematics — study of rotational motion without focusing on the forces or torques causing it.

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

Angular displacement (Δθ) — change in angular position.

Angular velocity (ω) — rate of change of angular position.

Initial angular velocity (ω_i) — angular velocity at the beginning of an interval.

Final angular velocity (ω_f) — angular velocity at the end of an interval.

Angular acceleration (α) — rate of change of angular velocity.

Constant angular acceleration — motion in which angular velocity changes by equal amounts during equal time intervals.

Radian (rad) — standard angular unit used in rotational equations.

Radian per second (rad/s) — SI unit of angular velocity.

Radian per second squared (rad/s²) — SI unit of angular acceleration.

Revolution — one complete rotation equal to 2π rad.

Tangential speed — linear speed of a point moving along a circular path.

Tangential acceleration — linear acceleration associated with changing angular speed.

Rolling without slipping — motion in which rotation and translation satisfy s = rθ, v = rω and a = rα.

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Key Takeaways

• Rotational kinematics uses angular quantities to describe rotational motion.
• The main variables are Δθ, ω_i, ω_f, α and t.
• Rotational equations closely parallel linear kinematic equations.
• The equations apply directly when angular acceleration is constant.
• ω_f = ω_i + αt relates angular velocity, acceleration and time.
• Δθ = ω_i t + 1/2 αt² relates angular displacement to time.
• ω_f² = ω_i² + 2αΔθ is useful when time is unknown.
• Δθ = [(ω_i + ω_f)/2]t uses average angular velocity.
• A good problem-solving method is to identify known and unknown variables before choosing an equation.
• Direction must be represented consistently using positive and negative signs.
• Angular displacement is measured in radians.
• Angular velocity is measured in rad/s.
• Angular acceleration is measured in rad/s².
• One revolution equals 2π rad.
• Rotational quantities connect to linear quantities through s = rθ, v = rω, and a_t = rα.
• Rolling wheels provide a direct connection between rotational and translational motion.
• The gradient of an ω-t graph gives angular acceleration.
• The area under an ω-t graph gives angular displacement.
• Real-world applications include wheels, motors, turbines, gears, pulleys and robotic systems.
• Rotational kinematics can predict starting motion, stopping motion and direction reversals.

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Check Your Understanding

1. Identify the symbols and SI units for angular displacement, angular velocity and angular acceleration.
2. Write the four main rotational kinematic equations for constant angular acceleration.
3. A wheel starts at ω_i = 5 rad/s and accelerates at 3 rad/s² for 4 s. Calculate its final angular velocity.
4. For Question 3, calculate the wheel's angular displacement.
5. A fan starts from rest and accelerates uniformly at 2.5 rad/s² for 8 s. Calculate its final angular velocity and angular displacement.
6. A disk slows from 40 rad/s to 10 rad/s in 6 s. Calculate its angular acceleration.
7. A wheel rotating at 30 rad/s slows uniformly at −5 rad/s². Calculate how long it takes to stop.
8. For Question 7, calculate the angular displacement before the wheel stops and convert it to revolutions.
9. A motor increases its angular velocity from 10 rad/s to 50 rad/s while rotating through 120 rad. Calculate its angular acceleration.
10. A wheel accelerates uniformly from 8 rad/s to 24 rad/s in 5 s. Calculate its angular displacement using average angular velocity.
11. A bicycle wheel has radius 0.35 m and angular velocity 18 rad/s. Calculate the bicycle's linear speed if the wheel rolls without slipping.
12. A car accelerates at 3.0 m/s². Its wheels have radius 0.30 m. Calculate their angular acceleration.
13. A wheel initially rotates counterclockwise at +18 rad/s and experiences constant angular acceleration −6 rad/s². Determine when it stops and describe its motion after that time if the acceleration continues.
14. An ω-t graph rises uniformly from 4 rad/s at t = 0 to 28 rad/s at t = 6 s. Determine the angular acceleration and angular displacement during the six seconds.
15. A rotating amusement ride starts from rest, reaches 12 rad/s in 8 s, and has radius 5 m. Assuming constant angular acceleration, calculate its angular acceleration, angular displacement, number of revolutions, and the final tangential speed of a rider at the outer edge.