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Circular Motion overview

Topic 1 of 3

Describe a turn

An angle tells you how far around a circle a body has moved. Its radius then connects that turn to the distance travelled and the speed along the path.

Radians measure arc length relative to radius

An arc of length sarc on a circle of radius r subtends an angle of magnitude θ, measured in radians:

θ = sarc/r
sarc = rθ
One complete turn: 2π rad = 360°

An angle of one radian cuts off an arc equal in length to the radius. The arc follows the circle; the straight chord between its endpoints is shorter. Use the radius from the rotation axis, not the diameter.

Radians are a ratio of two lengths, so the ratio is dimensionless. Keep the label rad to show that the number represents an angle. Convert degrees using θrad = θdegrees × π/180 before using sarc = rθ or the circular-motion equations.

Angular velocity, period and frequency

Angular velocity is the rate of change of angular displacement. In a stated plane, choose a viewing direction and positive rotation sense to give angular displacement a sign. For example, take anticlockwise viewed from above as positive; clockwise rotation then has negative angular velocity. An arc length remains non-negative.

Average angular velocity = Δθ/Δt

For uniform rotation, this equals the constant angular velocity ω. Its units are rad/s. Angular speed is its magnitude. In the speed equations below, v and ω denote non-negative speed and angular-speed magnitudes; the chosen rotation sense is stated separately.

The period T is the time for one complete revolution, in seconds. The frequency f is the number of revolutions per second, in hertz (Hz = s-1).

f = 1/T
ω = 2π/T = 2πf

Angular frequency is also written ω in rad/s: one cycle corresponds to 2π radians. A frequency in Hz is therefore not numerically equal to the angular rate in rad/s. Divide revolutions per minute by 60 to obtain revolutions per second before multiplying by 2π.

For a fixed radius, speed along the circle is arc distance divided by time. Substituting sarc = rθ gives:

v = sarc/t = rθ/t = rω

Instantaneous velocity is tangent to the circle. The curved arrow used to mark rotation sense is not a velocity vector pointing around the whole path.

Estimate the size first

A radius of about 0.4 m and a period of about 2 s give an angular rate of about 3 rad/s and speed of about 1 m/s. The inward acceleration is of order a few m/s2, so a roughly 0.2 kg marker would need an inward resultant of order 1 N.

These assumed rough scales are a check on magnitude. The acceleration and force pages explain why those quantities point inward and how to calculate them.

Worked uniform turn

Two markers on one rigid platform

A marker at radius 0.400 m completes one anticlockwise turn every 2.00 s, viewed from above. The rate is uniform.

f = 1/2.00 = 0.500 Hz = 30.0 revolutions/min
ω = 2π/2.00 = π rad/s
v = 0.400π = 1.26 m/s

In 0.750 s, starting from the radius pointing right:

Δθ = ωt = 0.750π = 3π/4 rad = 135°
sarc = rΔθ = 0.300π = 0.942 m

A turn measures an arc, not its chord

View the platform from above, with anticlockwise positive. The outer marker starts at the hollow point on the right; both markers advance through the same angle in 0.750 s.

Two markers share a 135-degree turn but have different arc distances and speedsA true top-view circle has centre O, outer radius 0.400 metres and an inner marker circle with half that radius, 0.200 metres. Both drawn circles use one spatial scale. From the outer rightmost starting point, the brown highlighted arc runs anticlockwise through 135 degrees, or three pi over four radians, to the upper-left outer marker. A dashed straight chord joins those endpoints but is not the arc distance. The inner marker ends on the same 135-degree radius. The outer arc distance is 0.942 metres to three significant figures. At the outer endpoint a separate blue velocity arrow is exactly tangent, pointing down and left, perpendicular to its radius. A curved brown arrow along the arc identifies the anticlockwise rotation sense. Velocity-arrow length has a different unit from the spatial scale.135° = 3π/4 radOuter arc: 0.942 mvO0.400 m0.200 mChord

Both markers have period 2.00 s and angular speed π rad/s. The outer speed is 1.26 m/s; the inner speed is 0.628 m/s. The brown arc measures distance travelled; the blue straight arrow shows instantaneous velocity.

The top view distinguishes radius, curved arc and straight chord. Both markers share the same angular displacement and period. The straight velocity arrow is tangent; the curved arrow gives the anticlockwise rotation sense.

A second marker at radius 0.200 m on the same rigid platform also takes 2.00 s per turn. Its angular speed is still π rad/s, but its tangential speed is 0.200π = 0.628 m/s, half the outer marker's. Equal period does not mean equal distance travelled.

Optional check Two markers at radii 0.200 m and 0.400 m share a rigid platform that turns uniformly once every 2.00 s. How do their angular speeds and tangential speeds compare?
Two markers at radii 0.200 m and 0.400 m share a rigid platform that turns uniformly once every 2.00 s. How do their angular speeds and tangential speeds compare?

Read the coordinates around a circle

Put the centre at the origin, with x to the right and y upwards on the top-view drawing. Measure θ anticlockwise from +x. Resolving the radius gives x = r cos θ and y = r sin θ. The radius r stays positive while the coordinates can be positive, zero or negative.

Position components depend on the angle

Measure x to the right and y upward from the circle's centre. The angle θ starts at the positive horizontal radius and increases anticlockwise. Then x/r = cos θ and y/r = sin θ.

x/r = cos θ

x divided by radius against angleFor a marker on a circle of radius r, the horizontal coordinate relative to its centre is x equals r cosine theta. Theta is measured anticlockwise from the positive horizontal radius and is the independent variable, in radians. The smooth normalized graph x over r has values positive one, zero, negative one, zero and positive one at angles zero, pi over two, pi, three pi over two and two pi. These are positions versus angle, not a time graph or a claim about simple harmonic motion.0π/2π3π/22π-10+1x/rAngle θ / rad

y/r = sin θ

y divided by radius against angleWith the same centre and angle convention, the upward coordinate is y equals r sine theta. Theta is the independent variable, in radians. The smooth normalized graph y over r has values zero, positive one, zero, negative one and zero at angles zero, pi over two, pi, three pi over two and two pi. The ordinates are dimensionless position ratios. No time scale or simple harmonic motion model is assigned.0π/2π3π/22π-10+1y/rAngle θ / rad

The vertical ratios are dimensionless. These curves plot position against angle; the horizontal axis is not time.

The horizontal axes show angle in radians from the right-pointing radius. The vertical axes show signed coordinates divided by the fixed radius. These are coordinate-versus-angle graphs, not speed or acceleration graphs.

At 0, π/2, π, 3π/2 and 2π, the marker passes right, top, left, bottom and right again. Accordingly, x/r follows 1, 0, -1, 0, 1 while y/r follows 0, 1, 0, -1, 0. These sine and cosine curves describe the coordinates of the circular position; the two-dimensional path is still a circle.

Small angles must be in radians

Near zero, the following useful approximations hold:

sin θ ≈ tan θ ≈ θ
cos θ ≈ 1
for small |θ|, with θ in radians

Here ≈ means approximately equal. At θ = 0.10 rad, sin θ = 0.0998334, tan θ = 0.1003347 and cos θ = 0.9950042. The approximations become closer as |θ| gets smaller; whether they are accurate enough depends on the required precision.

For r = 0.400 m and θ = 0.10 rad, the arc is rθ = 0.0400 m. The chord is 2r sin(θ/2) = 0.0399833 m, nearly the same length. This explains why a short chord can approximate a small arc. The earlier 135° turn is not a small angle; use the full trigonometric functions and arc relation there.

Measure repeated turns

On a secured rotating platform at a controlled steady rate, choose a visible reference line. Start and stop as the marker crosses that line in the same sense, counting a stated number of complete revolutions. Measure the radius from the axis to the marker's centre with a rule of suitable range and resolution.

Illustrative stopwatch readings for 20 complete turns are 39.8 s, 40.2 s and 40.0 s. Their mean is 40.0 s, giving T = 40.0/20 = 2.00 s. With radius 0.400 m, these values reproduce ω = π rad/s and v = 1.26 m/s. They are supplied example readings, not a claim that every platform turns at this rate.

Timing many turns reduces the fractional effect of start/stop timing. Repeats reveal scatter, but do not correct a wrongly located axis or a repeated miscount of whole turns. Check whether the rotation rate changes across the interval rather than assuming uniform motion from one average.

For an independent speed comparison, calibrated video can provide displacement over short known frame intervals. View perpendicular to the rotation plane, calibrate distance in that plane, identify the centre and verify the frame timing. A tilted camera distorts the geometry. A finite chord divided by time approximates arc speed; smaller angular intervals improve that approximation but must still be large enough to resolve position and time reliably.

Compare this independently estimated speed with rω obtained from the radius and a separately timed full revolution at the same steady rate. Calculating v from rω and substituting it back is not an independent experimental test of the relationship.