Circular Motion with Sine and Cosine

A point moving at constant speed around a circle has coordinates that are sine and cosine functions of time: x(t) = r cos(ωt + φ) and y(t) = r sin(ωt + φ), where r is the radius and ω is the angular velocity in radians per unit time. In brief: a wheel of radius 20 m turning once every 8 minutes has ω = 2π/8 = π/4 rad/min, and the height of a point on its rim is a sinusoid with amplitude 20 and period 8. The projection of circular motion onto a line is simple harmonic motion.

Circular motion links geometry and periodic functions. It explains why the graphs of sine and cosine describe wheels, planets, pendulums and alternating current, and it provides a method to build models for any periodic phenomenon. This guide shows how to write the coordinates of a point on a circle, how to compute speed and period, how to model a Ferris wheel and links the motion to the sinusoidal graphs.

How do you describe the position of a point on a circle?

For a circle with center (a, b) and radius r, a point at the angle θ (measured counterclockwise from the positive x-axis) has the coordinates (a + r cos θ, b + r sin θ). On the unit circle with center at the origin and r = 1, the point is (cos θ, sin θ).

If the angle changes uniformly with time, θ = ωt + φ, where φ is the starting angle at t = 0. Then x(t) = a + r cos(ωt + φ) and y(t) = b + r sin(ωt + φ). Time here plays the role of the input in the function, so the graph of y against t is a transformed sine.

What are angular velocity, period and speed?

• Period T: the time for one full revolution.
• Angular velocity ω = 2π/T (radians per time unit). If there are f revolutions per time unit, ω = 2πf.
• Linear speed v = rω: the distance travelled along the circle per time unit.
• Centripetal acceleration a = rω² = v²/r, directed to the center.

Example 1

A bicycle wheel of radius 0.35 m turns 3 times per second. Then ω = 2π · 3 = 6π ≈ 18.85 rad/s. A point on the rim travels at v = 0.35 · 18.85 ≈ 6.6 m/s ≈ 23.7 km/h, which is also the speed of the bicycle if the wheel rolls without slipping. The centripetal acceleration is 0.35 · 18.85² ≈ 124 m/s².

Example 2: the Earth

The Earth rotates once in about 24 hours, so ω = 2π/24 ≈ 0.2618 rad/h. At the equator the radius is about 6,378 km, so v ≈ 6378 · 0.2618 ≈ 1,670 km/h. At latitude 60°, the radius of the circle of latitude is 6378 cos 60° ≈ 3,189 km, and the speed is about 835 km/h.

How does circular motion give a sine wave?

If you look at a rotating point from the side, so that you see only its vertical (or horizontal) position, you see an oscillation up and down: y(t) = r sin(ωt + φ). This motion is called simple harmonic motion and is the motion of a mass on a spring and of a pendulum with small swings. The amplitude is r, the period is T = 2π/ω and the frequency is f = 1/T.

The velocity of this motion is the rate of change of y: v_y = rω cos(ωt + φ), which is the derivative of sine. The maximum speed rω occurs when the point crosses the midline.

Example 3: a pendulum

A pendulum swings through a horizontal distance of 10 cm either side of the center, with a period of 2 seconds. Then r = 10 cm, ω = π rad/s and x(t) = 10 sin(πt). At t = 0.5 s, x = 10 cm (the extreme). The speed at the center is rω = 10π ≈ 31.4 cm/s.

How do you model a Ferris wheel?

1. Determine the radius r from the diameter and the center height b: b = height of the lowest point + r.
2. Compute ω = 2π/T from the period.
3. Choose the starting angle φ from the position at t = 0.
4. Write h(t) = b + r sin(ωt + φ).

Example 4

A Ferris wheel has a diameter of 40 m, the lowest point is 2 m above the ground, and a rotation takes 8 minutes. A rider boards at the lowest point at t = 0. So r = 20, b = 22, ω = π/4. Starting at the lowest point means the sine begins at its minimum, which is the angle φ = −π/2: h(t) = 22 + 20 sin(πt/4 − π/2) = 22 − 20 cos(πt/4). Check: h(0) = 2 and h(4) = 42.

When is the rider above 32 m during the first rotation? 22 − 20 cos(πt/4) > 32 gives cos(πt/4) < −1/2, so 2π/3 < πt/4 < 4π/3 and 8/3 < t < 16/3 minutes. That is 8/3 minutes ≈ 2 min 40 s of each 8-minute turn.

Example 5: speed of the rider

The speed is v = rω = 20 · π/4 = 5π ≈ 15.7 m/min ≈ 0.26 m/s. The vertical speed is largest at the middle height (h = 22), where it equals the full speed 15.7 m/min.

Positions of the Ferris wheel over one rotation

Where do these ideas come from?

Greek astronomers described planetary motion with combinations of uniform circular motions, an idea used by Ptolemy in the second century. The link between circular motion and sinusoidal oscillation became central in the seventeenth century with the study of pendulums by Huygens and of springs by Hooke. In the nineteenth century Fourier’s analysis showed that any periodic motion can be built from circular motions, an idea that is still used in signal processing.

Where are they used today?

Alternating current is produced by a coil rotating in a magnetic field, so the voltage is V(t) = V₀ sin(ωt). Radio and sound waves are modelled with the same functions. Satellites in circular orbits have positions (r cos ωt, r sin ωt); for the International Space Station, an orbital period of about 92 minutes gives ω ≈ 0.0683 rad/min. Engineers design wheels, turbines and gears with the formulas for angular speed.

Which mistakes are common?

Mixing up the angular velocity and the speed

ω is in radians per time, and v = rω is a length per time. A larger radius means a larger speed for the same ω.

Using degrees in the formula for speed

v = rω works only with ω in radians per time unit.

Forgetting the midline

The height of a rider on a wheel is b + r sin(…), not r sin(…). The center of the wheel is above the ground.

Wrong starting phase

A rider starting at the bottom needs a negative cosine or a phase shift. Check h(0) against the problem.