Loci of Points on Moving Mechanisms
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Lesson overview
Constructing the path traced by a point on a mechanism such as a rolling circle or a linkage.
Go to lesson quiz →Exam guidance · WAEC WASSCE Paper 1 + Paper 2 Lens
Pacing
Objective speed on Paper 1 + structured step-by-step method marks on Paper 2 Theory/Practical.
How WAEC marks this
Examiners award method marks (M1) for stating the formula/definition and accuracy marks (A1) ONLY when SI units are written.
Watch out for
Writing bare numbers without working or omitting SI units loses up to 50% of Paper 2 marks even when the final value is right.
Lesson content
Overview — Loci of Points on Moving Mechanisms
Constructing the path traced by a point on a mechanism such as a rolling circle or a linkage. This lesson is at the advanced level — by the end, you'll be able to recognise this question type, apply a reliable method, and avoid the traps that cost marks.
Key Concept
A locus is the path traced by a point as it moves according to a defined rule or mechanism — for example, the path traced by a fixed point on the circumference of a circle as that circle rolls along a straight line (producing a curve called a cycloid) requires plotting the point's position at successive stages of the rolling motion.
Why It's Tested This Way
WAEC tests loci of moving mechanisms at the advanced level because constructing a locus requires combining geometric construction skill with an understanding of how a real mechanism actually moves, translating that motion into a series of accurately plotted points that are then joined into a smooth curve.
Method
When answering, divide the mechanism's motion into a manageable number of equal stages (commonly twelve equal divisions for a full rotation), construct the moving point's exact position at each individual stage, then join all the plotted points with a smooth curve to complete the locus.
Worked Example
Worked example: To construct the cycloid traced by a point on a circle of given radius rolling along a straight line, divide the circle's circumference into twelve equal arcs, mark the corresponding twelve equal divisions along the straight line the circle rolls along, then plot the tracing point's position at each of the twelve stages by rotating the circle progressively and marking where the original point has moved to.
Second Worked Example
Second worked example: The locus of a point on a connecting rod within a simple slider-crank mechanism (as found in an engine) is constructed by plotting the rod's specific point's position at successive angles of the crank's rotation, again typically at twelve equal angular divisions, before joining the plotted points into the resulting curve.
Common Traps
Common trap: plotting too few stages of the mechanism's motion (using only three or four positions instead of a finer division like twelve), which produces an inaccurate, poorly-shaped curve when the points are joined — WAEC specifically expects a sufficient number of accurately plotted points for a smooth, correct locus.
Lesson Summary
Remember: (1) identify what is being asked, (2) apply the WAEC strategy, (3) check against the traps. Complete the 10-question quiz below to unlock the next lesson — you need at least 60% to pass.
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