Describing Circular Motion: Angle, Period and Frequency – Uniform Circular Motion (Senior 3 Physics) | Ganzaa

Uniform Circular Motion

Lessons

Senior 3 · Physics

Uniform Circular Motion

4 lessons

Lesson 1 of 4

Describing Circular Motion: Angle, Period and Frequency

25 min

The valve that keeps coming back

In a bicycle repair shop, a mechanic turns a bicycle upside down, lifts the chain back onto its gear and turns the pedal. The back wheel spins. Watch the valve, where the air goes in: it goes round, passes the frame, and comes back to the same place again and again.

In S1 and S2 you described motion along a straight line with distance, displacement, speed and acceleration. Those words do not work well here. After one full turn the valve is back where it started, so its displacement is zero, yet it has clearly been moving. To describe something that goes round, physicists measure angles and times for one turn. This lesson introduces those quantities.

Definition

Circular motion is the motion of a body along a circular path (trajectory). The fixed point at the middle is the centre, and the distance from the centre to the body is the radius $r$ .

Uniform circular motion is circular motion at a constant speed: the body covers equal lengths of the circle in equal times. Examples: a point on a steadily turning fan blade; the tip of a clock's second hand.

Experiment 1: timing a turning wheel

You need: a bicycle, a piece of coloured tape, a metre rule, a stopwatch, and a pointer (a stick taped to a stool or a clamp stand).

Turn the bicycle upside down so that it rests on its saddle and handlebars.
Stick the tape on the rim beside the valve. Fix the pointer so that it points at the top of the back wheel.
Measure the radius $r$ of the wheel, from the centre of the hub to the outside of the tyre.
Turn the pedal by hand at a steady rate. When the tape passes the pointer, start the stopwatch and count "zero".
Count 20 passes of the tape, stop the watch, and record the time. Repeat twice and take the mean.

Figure 1.1: Turning the pedal steadily keeps the back wheel turning at a constant rate. Each pass of the tape under the pointer is one complete turn (one revolution).

Trial	Time for 20 turns (s)
1	16.2
2	15.8
3	16.0
Mean	16.0

These are illustrative results from one class. Your own times will depend on how fast you turn the pedal. Timing 20 turns rather than one makes the reaction-time error of the stopwatch 20 times smaller for each turn.

Important

Safety. Keep fingers, loose sleeves, long hair and scarves away from the spokes, the chain and the gear wheel. Never stop the wheel with your hand: stop turning the pedal and squeeze the brake. Work on a flat floor with the bicycle held steady by a partner.

Measuring angles in radians

As the wheel turns, a line from the hub to the valve sweeps out an angle. In circular motion we measure this angular displacement $\theta$ (the Greek letter theta) in radians, not degrees, because radians link the angle directly to the distance moved along the circle.

Figure 1.2: (a) When a point moves from A to B along an arc of length $s$ , the radius sweeps out an angle $\theta = \frac{s}{r}$ . (b) When the arc is exactly as long as the radius, the angle is one radian.

For one complete turn, the arc is the whole circumference $2\pi r$ , so

\theta = \frac{2\pi r}{r} = 2\pi\ \text{rad}

2\pi\ \text{rad} = 360^\circ, \quad 1\ \text{rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ

Degrees	30°	90°	180°	270°	360°
Radians	$\frac{\pi}{6}$	$\frac{\pi}{2}$	$π \pi$	$\frac{3\pi}{2}$	$2\pi$

Definition

Angular displacement ( $\theta$ ) is the angle swept out at the centre by the line joining the centre to the moving body. SI unit: the radian (rad).

One radian is the angle at the centre of a circle made by an arc whose length equals the radius. In general $\theta = \frac{s}{r}$ , where $s$ is the arc length. Example: a point moving 0.70 m round a wheel of radius 0.35 m turns through $\theta = \frac{0.70}{0.35} = 2.0\ \text{rad}$ .

Period, frequency and angular velocity

From Experiment 1, the wheel made 20 turns in 16.0 s. Three quantities describe how fast it turns.

The period $T$ is the time for one complete turn: $T = \frac{16.0\ \text{s}}{20} = 0.80\ \text{s}$ .
The frequency $f$ is the number of turns each second: $f = \frac{20}{16.0\ \text{s}} = 1.25\ \text{Hz}$ . One hertz (Hz) is one turn per second, $1\ \mathrm{s^{-1}}$ .
The angular velocity $\omega$ (omega) is the angle turned each second.

The period and frequency are reciprocals: if one turn takes 0.80 s, then $\frac{1}{0.80} = 1.25$ turns happen each second.

f = \frac{1}{T}

T = \frac{1}{f}

Angular velocity is the angular displacement divided by the time taken:

\omega = \frac{\Delta\theta}{\Delta t}

In one period the body turns through $2\pi$ rad, so for uniform circular motion

\omega = \frac{2\pi}{T} = 2\pi f

For the bicycle wheel: $\omega = \frac{2\pi}{0.80\ \text{s}} = 7.9\ \mathrm{rad\,s^{-1}}$ . In uniform circular motion, $\omega$ , $T$ and $f$ all stay constant.

Definition

Period ( $T$ ): the time for one complete revolution. SI unit: the second (s).

Frequency ( $f$ ): the number of revolutions per second, $f = \frac{1}{T}$ . SI unit: the hertz (Hz), where $1\ \text{Hz} = 1\ \mathrm{s^{-1}}$ . Machines are often rated in revolutions per minute (rev/min, or rpm): divide by 60 to get hertz.

Angular velocity ( $\omega$ ): the rate of change of angular displacement, $\omega = \frac{\Delta\theta}{\Delta t} = \frac{2\pi}{T} = 2\pi f$ . SI unit: the radian per second ( $\mathrm{rad\,s^{-1}}$ ).

Worked example 1: the hands of a clock

Find the angular velocity of (a) the second hand and (b) the minute hand of a clock.

(a) The second hand makes one turn in 60 s, so $T = 60\ \text{s}$ :

\omega = \frac{2\pi}{T} = \frac{2\pi}{60\ \text{s}} = 0.105\ \mathrm{rad\,s^{-1}}

(b) The minute hand makes one turn in one hour, $T = 3600\ \text{s}$ :

\omega = \frac{2\pi}{3600\ \text{s}} = 1.75 \times 10^{-3}\ \mathrm{rad\,s^{-1}}

The minute hand turns 60 times more slowly, so its angular velocity is 60 times smaller.

Worked example 2: a maize mill motor

The electric motor of a maize mill is rated at 1450 rev/min. Find (a) its frequency, (b) its angular velocity, and (c) the angle it turns through in 0.50 s.

(a) $f = \frac{1450}{60\ \text{s}} = 24.2\ \text{Hz}$

(b) $\omega = 2\pi f = 2\pi \times 24.17\ \text{Hz} = 152\ \mathrm{rad\,s^{-1}}$

(c) $\omega$ is constant, so $\theta = \omega t = 151.8\ \mathrm{rad\,s^{-1}} \times 0.50\ \text{s} = 76\ \text{rad}$ . That is $\frac{76}{2\pi} \approx 12$ complete turns in half a second.

Worked example 3: the turning Earth

The Earth turns once on its axis in about 24 hours. Find its angular velocity.

T = 24 \times 3600\ \text{s} = 86\,400\ \text{s}

\omega = \frac{2\pi}{86\,400\ \text{s}} = 7.27 \times 10^{-5}\ \mathrm{rad\,s^{-1}}

Every place on Earth, Kigali included, turns with this same angular velocity.

Warning

Radians, not degrees; and $\omega$ is not $f$ . The formula $\omega = 2\pi f$ gives $\omega$ in $\mathrm{rad\,s^{-1}}$ only because $2\pi$ rad is one turn. If you use 360 instead of $2\pi$ , you get degrees per second, which does not fit the formulas in the next lessons. A wheel with $f = 2\ \text{Hz}$ has $\omega = 4\pi = 12.6\ \mathrm{rad\,s^{-1}}$ , not $2\ \mathrm{rad\,s^{-1}}$ .

Tip

One turn, three ways: one revolution = $2\pi$ rad = 360°, and it takes one period $T$ . So $\omega = \frac{\text{one turn in radians}}{\text{time for one turn}} = \frac{2\pi}{T}$ . To convert rev/min to $\mathrm{rad\,s^{-1}}$ , multiply by $\frac{2\pi}{60}$ .

Connecting period and frequency to angular velocity (Khan Academy). Watch how the angle of one full turn, 2π radians, links the period to the angular velocity.

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Practice

Define (a) uniform circular motion; (b) period; (c) frequency.
Convert to radians: (a) 180°; (b) 60°; (c) 450°. Leave your answers in terms of $π \pi$ .
Convert 2.5 rad to degrees.
A point on a wheel of radius 0.30 m moves 0.90 m along the rim. Through what angle, in radians, has the wheel turned?
A wheel makes 45 turns in 30 s. Find its period, its frequency and its angular velocity.
A ceiling fan turns at 900 rev/min. Find its angular velocity in $\mathrm{rad\,s^{-1}}$ .
A turntable has an angular velocity of $5.0\ \mathrm{rad\,s^{-1}}$ . Through what angle does it turn in 4.0 s? How many complete turns is that?
A pupil times 10 turns of a wheel as 7.0 s and says that the period is 7.0 s and the frequency 0.14 Hz. Find her mistake and give the correct values.

Show answersHide answers

(a) Motion in a circle at constant speed. (b) The time for one complete revolution. (c) The number of revolutions per second.
(a) $π \pi$ rad; (b) $\frac{\pi}{3}$ rad; (c) $\frac{5\pi}{2}$ rad.
$2.5 \times \frac{180^\circ}{\pi} = 143^\circ$
$\theta = \frac{s}{r} = \frac{0.90}{0.30} = 3.0\ \text{rad}$
$T = \frac{30}{45} = 0.67\ \text{s}$ ; $f = \frac{45}{30} = 1.5\ \text{Hz}$ ; $\omega = 2\pi \times 1.5 = 9.4\ \mathrm{rad\,s^{-1}}$ .
$f = \frac{900}{60} = 15\ \text{Hz}$ , so $\omega = 2\pi \times 15 = 94\ \mathrm{rad\,s^{-1}}$ .
$\theta = \omega t = 5.0 \times 4.0 = 20\ \text{rad}$ , which is $\frac{20}{2\pi} = 3.2$ turns: 3 complete turns.
7.0 s is the time for 10 turns, not one. $T = \frac{7.0}{10} = 0.70\ \text{s}$ and $f = \frac{1}{0.70} = 1.4\ \text{Hz}$ .

Key points

Uniform circular motion is motion round a circle at constant speed.
Angles are measured in radians: $\theta = \frac{s}{r}$ , and one turn is $2\pi$ rad = 360°.
Period $T$ (s) is the time for one turn; frequency $f = \frac{1}{T}$ (Hz) is turns per second.
Angular velocity $\omega = \frac{\Delta\theta}{\Delta t} = \frac{2\pi}{T} = 2\pi f$ , in $\mathrm{rad\,s^{-1}}$ .

Sources and credits

6.1: Rotation Angle and Angular Velocity, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Consulted for the physics; no text was copied.
Photo: Bicycle wheel (close-up).JPG by Vegasjon, Wikimedia Commons, licensed under CC BY-SA 4.0.
Video: Connecting period and frequency to angular velocity | AP Physics 1 | Khan Academy, Khan Academy, embedded from YouTube.

Written by Ganzaa: the lesson text, the experiment and its illustrative results, the diagrams, worked examples and practice exercises.

1The valve that keeps coming back

2Experiment 1: timing a turning wheel

3Measuring angles in radians

4Period, frequency and angular velocity

The valve that keeps coming back

Experiment 1: timing a turning wheel

Measuring angles in radians

Period, frequency and angular velocity