Simple Harmonic Motion

≈ 42 min

Simple Harmonic Motion

Uniform motion has abrupt velocity changes at each end of the rise. Simple harmonic motion (SHM) removes them: the follower starts from rest, accelerates smoothly to maximum speed at mid-travel, then decelerates smoothly back to rest.

The construction is geometric, not calculated. Draw a semicircle whose diameter equals the total rise. Divide the semicircle into the same number of equal angular parts as the rotation divisions. Project each point horizontally onto its rotation ordinate. The resulting displacement curve is a sine curve.

Why the divisions are angular. Equal angles around the semicircle project to unequal vertical spacings — closely spaced near the ends and widely spaced in the middle. That uneven projection is precisely what produces the smooth acceleration and deceleration.

Worked example. With six divisions the first rise step is small, the middle steps are large, and the last is small again. A learner who divides the semicircle's diameter into six equal parts instead of its arc gets equal steps back — which is uniform motion drawn the long way round.

Core checkpoint: Divide the semicircle by angle, not the diameter by length. The unequal projection is the whole point.

How is the semicircle divided when constructing simple harmonic motion?

Into equal angular parts around its arc.

Into equal lengths along its diameter.

Into parts proportional to the rotation angle.

It is not divided at all.

Where is the follower moving fastest in simple harmonic motion?

At mid-travel.

At the very start of the rise.

At the very end of the rise.

The speed is constant throughout.

What advantage does SHM have over uniform motion?

The follower starts and ends at rest with smooth acceleration, removing the shock loading of uniform motion's abrupt corners.

The follower moves further for the same rotation.

The cam is smaller for the same rise.

It is easier to construct.

A learner divides the semicircle's diameter into equal lengths instead of its arc into equal angles. What motion results?

Uniform motion — the equal steps return, reintroducing the shock loading SHM was meant to remove.

A faster version of SHM.

No motion can be plotted.

Uniform acceleration.