AP® Physics 1: Algebra-Based · Unit 7 · Topic 7.3
Simple Harmonic Motion as the Shadow of a Circle
Simple harmonic motion is what you see when you project uniform circular motion onto a single axis: the shadow of a point moving steadily around a circle swings back and forth exactly like a spring or pendulum.
Picture a ball glued to the edge of a turntable, spinning at constant speed, lit from the side so it casts a shadow on a wall. The ball's actual path is a circle, but the shadow only moves along one line, left and right. As the ball goes around, the shadow slows down, stops, reverses, speeds up through the middle, and slows again approaching the other side. That motion is simple harmonic motion, and the position of the shadow over time traces a perfect sine curve.
The math behind this is projection. If the ball sits at angle theta = omega t around a circle of radius A, its horizontal position is x = A cos(omega t). That is exactly the position function of an oscillator with amplitude A and angular frequency omega. Velocity and acceleration come the same way: the ball's constant tangential speed around the circle is A omega, and this equals the oscillator's maximum speed, reached only when the ball's velocity vector points fully along the viewed axis. That happens when the ball crosses the axis itself, which is the shadow's center. At the far edges of the swing, the ball's velocity points perpendicular to the axis, so the shadow's speed there drops to zero even though the ball never stops moving.
This picture is why period in SHM does not depend on amplitude: a bigger circle just means the ball travels farther, but at a proportionally faster tangential speed, so going around still takes the same time 2 π / omega. It also ties oscillations directly to rotational motion, reusing angular velocity, angular position, and centripetal acceleration to describe back-and-forth motion without inventing new tools.
Key terms
- Uniform circular motion
- Motion around a circle at constant speed, used here as the hidden full motion behind an SHM shadow.
- Angular frequency (omega)
- The rate of change of phase in SHM, matched to the angular velocity of the reference circular motion.
- Projection
- Taking only one coordinate, such as x, of a point's position as it moves through two dimensions.
- Amplitude (A)
- The maximum displacement of the oscillator from center, equal to the radius of the reference circle.
- Tangential speed
- The constant speed of the object moving around the circle, equal to A omega and to the oscillator's maximum speed.
Watch it explained
- How it worksHow does projecting uniform circular motion onto one axis produce a perfect sine in time?
- How it worksWalk me through matching the turntable's angular velocity to the oscillator's angular frequency.
- Going furtherWhat does the circular picture reveal about why maximum speed happens at the center?
- Ties togetherHow does this circle picture connect oscillations back to the rotational motion unit?
Check yourself
End of topic.