SHM as a Shadow of Circular Motion

Watch a point go round a circle from the side, and it bobs in SHM.

8 minUpdated 21 October 2026reference circleprojectionphase differenceuniform circular motion

Lesson at a glanceShow the key points

A point moving uniformly on a circle of radius A at angular speed ω has x = A cos ωt and y = A sin ωt (12.11, 12.12).
Each projection is SHM with amplitude A and angular frequency ω; the two differ in phase by π/2.
The phase of the SHM is the angle of the rotating radius; the period is the time for one revolution.

Shine a light from the side at a ball going round a turntable and watch its shadow on the wall. The shadow bobs back and forth in perfect simple harmonic motion.

The projection

Let a point P move round a circle of radius A at constant angular speed ω, starting on the x-axis at t = 0. Take the centre as the origin. At time t the radius OP has turned through ωt, so the feet of the perpendiculars from P onto the axes are at

Projections of uniform circular motion(12.11, 12.12)

x = A\cos\omega t, \qquad y = A\sin\omega t

Each is exactly the equation of SHM, with amplitude A (the circle’s radius) and angular frequency ω (the angular speed). The two shadows are a quarter-cycle apart, because $\cos\omega t = \sin(\omega t + \pi/2)$ .

Interactive

SHM is circular motion seen edge-on

A point moves steadily round a circle. Its shadow on the vertical diameter (gold) bobs up and down in SHM, tracing a sine curve against time on the right. The shadow on the horizontal diameter (blue) does the same, a quarter-cycle ahead.

\theta = \omega t

0.00π

x = A\cos\omega t

1.00

y = A\sin\omega t

0.00

Angular speed

\omega

1.50rad/s

What to remember

1The projection of uniform circular motion onto any diameter is SHM.
2Amplitude = circle radius; angular frequency = angular speed; phase = angle of the radius.
3Projections on two perpendicular diameters are SHMs differing in phase by π/2.

Check yourself

A point goes round a circle of radius 10 cm twice a second. Its shadow on a diameter has period…

When the shadow is at the end of its swing, the rotating point is…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadComparing simple harmonic motion and circular motionOpenStax University Physics §15.3 ReadSHM: position, velocity, accelerationHyperPhysics

PreviousThe Equation of SHM Next Energy in SHM

SHM as a Shadow of Circular Motion

Watch a point go round a circle from the side, and it bobs in SHM.

8 minUpdated 21 October 2026reference circleprojectionphase differenceuniform circular motion

Lesson at a glanceShow the key points

A point moving uniformly on a circle of radius A at angular speed ω has x = A cos ωt and y = A sin ωt (12.11, 12.12).
Each projection is SHM with amplitude A and angular frequency ω; the two differ in phase by π/2.
The phase of the SHM is the angle of the rotating radius; the period is the time for one revolution.

The projection

Projections of uniform circular motion(12.11, 12.12)

x = A\cos\omega t, \qquad y = A\sin\omega t

Each is exactly the equation of SHM, with amplitude A (the circle’s radius) and angular frequency ω (the angular speed). The two shadows are a quarter-cycle apart, because

\cos\omega t = \sin(\omega t + \pi/2)

Interactive

SHM is circular motion seen edge-on

\theta = \omega t

0.00π

x = A\cos\omega t

1.00

y = A\sin\omega t

0.00

Angular speed

\omega

1.50rad/s

What to remember

1The projection of uniform circular motion onto any diameter is SHM.
2Amplitude = circle radius; angular frequency = angular speed; phase = angle of the radius.
3Projections on two perpendicular diameters are SHMs differing in phase by π/2.

Check yourself

A point goes round a circle of radius 10 cm twice a second. Its shadow on a diameter has period…

When the shadow is at the end of its swing, the rotating point is…

The projection#

Check yourself

Keep exploring

The projection#

Check yourself

Keep exploring

The projection

The projection