For example, at:

f = 5 Hz

the angular velocity is:

ω = 2π(5)

ω = 10π rad/s

≈ 31.4 rad/s

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Period and Angular Velocity

The period, T, is the time required for one complete revolution.

Frequency and period are related by:

f = 1/T

Therefore:

ω = 2π/T

This relationship is extremely useful in circular and rotational motion.

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Worked Example: Using Period

A rotating platform completes one revolution every:

T = 4 s

Then:

ω = 2π/4

ω = π/2 rad/s

≈ 1.57 rad/s

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Comparing Frequency and Angular Velocity

They are connected by:

f = 1/T

and:

ω = 2πf = 2π/T

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Revolutions Per Minute

Rotating machinery is often described using:

rpm = revolutions per minute

To convert rpm to rad/s:

1. convert revolutions to radians
2. convert minutes to seconds

Since:

1 rev = 2π rad

and:

1 min = 60 s

then:

ω = rpm × 2π/60

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Worked Example: 120 rpm

A motor rotates at:

120 rpm

Convert to revolutions per second:

120/60 = 2 rev/s

Therefore:

f = 2 Hz

Then:

ω = 2π(2)

ω = 4π rad/s

≈ 12.6 rad/s

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Worked Example: 3000 rpm

A motor rotates at:

3000 rpm

First:

f = 3000/60

f = 50 Hz

Then:

ω = 2π(50)

ω = 100π rad/s

≈ 314 rad/s

High-speed machinery can therefore have very large angular velocities.

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Angular Velocity and Linear Velocity

Consider a point on the edge of a rotating wheel.

Its angular displacement is related to its arc length by:

s = rθ

Divide both sides by time:

s/t = r(θ/t)

Since:

v = s/t

and:

ω = θ/t

we obtain:

v = rω

This is one of the most important relationships in rotational motion.

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Tangential Velocity

The linear velocity of a point moving around a circular path is often called its tangential velocity.

Its magnitude is:

v = rω

The direction of the velocity is:

tangent to the circular path

at that instant.

Angular velocity describes the rotation of the entire rigid object.

Tangential velocity describes the linear motion of a particular point on that object.

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Same Angular Velocity, Different Linear Speeds

Every point on a rigid rotating disk has the same angular velocity.

However:

v = rω

so points farther from the axis have greater linear speeds.

Suppose a disk rotates at:

ω = 6 rad/s

Point A is:

r_A = 0.10 m

from the centre.

Point B is:

r_B = 0.40 m

Then:

v_A = (0.10)(6)

v_A = 0.60 m/s

and:

v_B = (0.40)(6)

v_B = 2.4 m/s

Both points rotate at 6 rad/s, but Point B moves four times faster linearly.

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Why the Outer Edge Moves Faster

During the same time interval, all points rotate through the same angle.

However, an outer point travels along a larger circle.

Therefore, it must cover a greater distance in the same time.

This means:

greater radius → greater tangential speed

when angular velocity is the same.

---

Angular Velocity Versus Linear Velocity

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Worked Example: Rotating Wheel

A wheel has radius:

0.50 m

and rotates at:

ω = 12 rad/s

Find the tangential speed at its edge.

Use:

v = rω

v = (0.50)(12)

v = 6.0 m/s

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Worked Example: Finding Angular Velocity

A point on the edge of a rotating disk moves at:

v = 8.0 m/s

The disk has radius:

r = 0.40 m

Use:

ω = v/r

ω = 8.0/0.40

ω = 20 rad/s

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Rolling Wheels

For a wheel rolling without slipping:

v = rω

where v is also the linear speed of the wheel's centre relative to the ground.

This connects the rotation of a wheel with the forward motion of a vehicle.

---

Worked Example: Bicycle Wheel

A bicycle wheel has radius:

0.35 m

and angular velocity:

ω = 15 rad/s

Then:

v = rω

v = (0.35)(15)

v = 5.25 m/s

Convert to km/h:

5.25 × 3.6

= 18.9 km/h

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Car Wheel Example

A car travels at:

27 m/s

Its wheel radius is:

0.30 m

Assuming no slipping:

ω = v/r

ω = 27/0.30

ω = 90 rad/s

Its frequency is:

f = ω/(2π)

f ≈ 14.3 Hz

Therefore, each wheel completes about:

14.3 revolutions per second

or:

14.3 × 60 ≈ 860 rpm

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Angular Velocity of a Clock Hand

The minute hand completes one clockwise revolution every:

3600 s

Therefore:

ω = −2π/3600

ω ≈ −1.75 × 10⁻³ rad/s

The negative sign represents clockwise rotation.

---

Hour Hand Angular Velocity

The hour hand completes one revolution every:

12 hours

Convert:

12 h = 43,200 s

Therefore:

ω = −2π/43,200

ω ≈ −1.45 × 10⁻⁴ rad/s

The minute hand therefore has a much greater angular speed than the hour hand.

---

Angular Velocity in Gears

When two gears mesh without slipping, their edges have the same tangential speed:

v₁ = v₂

Since:

v = rω

then:

r₁ω₁ = r₂ω₂

Therefore:

ω₂/ω₁ = r₁/r₂

A smaller gear rotates faster than a larger gear.

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Worked Gear Example

Gear A has radius:

r_A = 4 cm

and rotates at:

ω_A = 30 rad/s

Gear B has radius:

r_B = 12 cm

Then:

r_A ω_A = r_B ω_B

(4)(30) = (12)ω_B

ω_B = 10 rad/s

Because meshing gears rotate in opposite directions, if:

ω_A = +30 rad/s

then:

ω_B = −10 rad/s

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Angular Velocity in Rotating Machinery

Angular velocity is important in:

• motors
• turbines
• generators
• fans
• drills
• engines
• washing machines
• hard drives
• industrial machinery

Machines often have specified rotational speeds in rpm.

Engineers convert these values into rad/s when performing calculations involving rotational motion.

---

Angular Velocity in Astronomy

Angular velocity also appears in astronomy.

Earth rotates once approximately every 24 hours.

Its average angular velocity is approximately:

ω = 2π/(24 × 3600)

ω ≈ 7.27 × 10⁻⁵ rad/s

This angular velocity affects:

• day and night
• apparent motion of stars
• launch trajectories
• motion of points on Earth's surface

---

Different Linear Speeds on Earth

All locations on Earth share approximately the same angular velocity due to Earth's rotation.

But their linear speeds are different.

A point near the equator travels around a much larger circular path than a point near the poles.

Therefore:

equatorial regions have greater linear rotational speed

while:

near the poles, linear rotational speed is much smaller.

This is another example of:

v = rω

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Angular Velocity as a Vector

In more advanced physics, angular velocity is treated as a vector.

Its direction lies along the axis of rotation, rather than tangent to the circular path.

The direction is determined using the:

right-hand rule.

Curl the fingers of your right hand in the direction of rotation.

Your thumb points in the direction of the angular velocity vector.

For counterclockwise rotation viewed from above, the angular velocity vector points upward from the plane.

For clockwise rotation, it points in the opposite direction.

For many introductory problems, using positive and negative signs is sufficient.

---

Instantaneous Angular Velocity from Data

Suppose angular position data are:

Average angular velocity from t = 2 s to t = 4 s is:

ω_avg = (20 − 6)/(4 − 2)

ω_avg = 14/2

ω_avg = 7 rad/s

The increasing changes in θ also show that the object is rotating faster as time passes.

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Estimating Instantaneous Angular Velocity

Suppose we want the angular velocity at approximately:

t = 2 s

and have nearby data:

θ(1.9 s) = 5.61 rad

θ(2.1 s) = 6.41 rad

Approximate the instantaneous angular velocity using the small interval:

ω ≈ Δθ/Δt

ω ≈ (6.41 − 5.61)/(2.1 − 1.9)

ω ≈ 0.80/0.20

ω ≈ 4.0 rad/s

Using data close to the desired instant gives an estimate of instantaneous angular velocity.

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Reversing Direction

Angular velocity can change sign.

Suppose:

• initially ω = +5 rad/s
• later ω = 0
• finally ω = −3 rad/s

The object:

1. initially rotates counterclockwise
2. slows to a stop
3. reverses direction
4. rotates clockwise

Therefore, the sign of angular velocity contains important information about rotational direction.

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Angular Velocity and Centripetal Acceleration

For circular motion:

a_c = v²/r

Since:

v = rω

substitute:

a_c = (rω)²/r

Therefore:

a_c = rω²

This connects angular velocity to centripetal acceleration.

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Worked Example: Centripetal Acceleration

A point is:

0.50 m

from the centre of a rotating disk.

The disk rotates at:

ω = 8 rad/s

Then:

a_c = rω²

a_c = (0.50)(8²)

a_c = (0.50)(64)

a_c = 32 m/s²

Even constant angular velocity produces centripetal acceleration because the direction of linear velocity continually changes.

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Constant Angular Velocity Does Not Mean Zero Acceleration

This is an important distinction.

If:

ω = constant

then angular velocity is not changing.

However, a point on the rotating object is continuously changing the direction of its linear velocity.

Therefore, it has centripetal acceleration toward the centre.

So:

constant angular velocity ≠ zero linear acceleration

for a point undergoing circular motion.

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A Rotational Motion Comparison

Many linear-motion ideas have rotational equivalents:

These parallels make rotational motion easier to analyze once linear kinematics is understood.

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Problem-Solving Strategy

When solving an angular velocity problem:

Identify the known quantities

Look for:

• θ
• Δθ
• time
• number of revolutions
• frequency
• period
• radius
• linear speed
• rpm

Convert units if necessary

Remember:

1 rev = 2π rad

1 min = 60 s

180° = π rad

Choose the relationship

Average angular velocity:

ω_avg = Δθ/Δt

Constant angular velocity:

Δθ = ωt

Frequency:

ω = 2πf

Period:

ω = 2π/T

Linear speed:

v = rω

Then check whether the sign and units make sense.

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Common Mistakes

Confusing angular velocity with linear velocity

Angular velocity is measured in rad/s; linear velocity is measured in m/s.

Forgetting direction

Angular velocity can be positive or negative.

Using degrees directly when rad/s is required

Convert degrees to radians before using standard rotational equations.

Forgetting that one revolution is 2π rad

It is not π rad.

Using rpm as though it were rad/s

rpm must be converted.

Thinking all points on a rotating disk have the same linear speed

They have the same angular velocity, but linear speed increases with radius.

Thinking a point at the centre has a large linear speed

At r = 0:

v = rω = 0

even though the disk has nonzero angular velocity.

Thinking constant angular velocity means no acceleration

Points moving in circles still have centripetal acceleration.

Ignoring the sign when rotation reverses

A change from positive to negative ω indicates a reversal of rotational direction.

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Key Terms

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

Average angular velocity — angular displacement divided by elapsed time.

Instantaneous angular velocity — rate of change of angular position at a particular instant.

Angular speed — magnitude of angular velocity.

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

Frequency (f) — number of complete revolutions or cycles per second.

Hertz (Hz) — unit of frequency equal to one cycle per second.

Period (T) — time required for one complete revolution.

Revolution per minute (rpm) — common unit of rotational speed.

Tangential velocity — instantaneous linear velocity tangent to a circular path.

Rigid body — object whose points remain fixed relative to one another.

Centripetal acceleration — inward acceleration required for circular motion.

Right-hand rule — method used to determine the vector direction of angular velocity.

Axis of rotation — line around which an object rotates.

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Key Takeaways

• Angular velocity describes how rapidly angular position changes.
• Angular velocity is represented by ω.
• The SI unit of angular velocity is rad/s.
• Average angular velocity is ω_avg = Δθ/Δt.
• Instantaneous angular velocity is the rate of change of θ at a particular instant.
• The gradient of an angular position-time graph represents angular velocity.
• Counterclockwise rotation is usually positive.
• Clockwise rotation is usually negative.
• Angular speed is the magnitude of angular velocity.
• One complete revolution equals 2π rad.
• Frequency and angular velocity are related by ω = 2πf.
• Period and angular velocity are related by ω = 2π/T.
• rpm can be converted to rad/s using ω = rpm × 2π/60.
• Angular and linear velocity are connected by v = rω.
• All points on a rigid rotating object have the same angular velocity.
• Points farther from the axis have greater tangential speeds.
• Rolling without slipping connects wheel rotation to vehicle speed through v = rω.
• Meshing gears have equal tangential speeds at their contact point but different angular velocities if their radii differ.
• Angular velocity can change sign when an object reverses rotational direction.
• A point moving with constant angular velocity can still have centripetal acceleration.
• Centripetal acceleration can be written as a_c = rω².

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Check Your Understanding

1. Define angular velocity and state its SI unit.
2. Explain the difference between angular speed and angular velocity.
3. A wheel rotates through 24 rad in 6 s. Calculate its average angular velocity.
4. A disk rotates clockwise through 15 rad in 3 s. Calculate its average angular velocity using the usual sign convention.
5. A motor rotates at a constant angular velocity of 8 rad/s for 12 s. Calculate its angular displacement.
6. A wheel completes 30 revolutions in 5 s. Calculate its frequency and angular velocity.
7. Convert 600 rpm to rad/s.
8. A rotating platform has a period of 2.5 s. Calculate its angular velocity.
9. A disk rotates at 10 rad/s. Calculate the tangential speed of a point 0.40 m from its centre.
10. Two points are located 0.20 m and 0.80 m from the centre of a rigid disk rotating at 5 rad/s. Compare their angular velocities and calculate their tangential speeds.
11. A bicycle travels at 7.0 m/s. Its wheels have radius 0.35 m. Assuming no slipping, calculate the angular velocity of the wheels.
12. Gear A has radius 5 cm and rotates at 24 rad/s. It drives Gear B of radius 15 cm. Calculate Gear B's angular velocity, including its direction relative to Gear A.
13. An angular position-time graph has a gradient of −6 rad/s. Explain what this tells you about the object's rotational motion.
14. A point 0.25 m from the centre of a disk rotates at 12 rad/s. Calculate its tangential speed and centripetal acceleration.
15. A student says, "Every point on a spinning wheel has the same velocity because the whole wheel has the same angular velocity." Evaluate this statement, clearly distinguishing between angular velocity and linear velocity.