Composite shape freely suspended from a point, hanging in equilibrium at an angle to the vertical
IAL M2 · Centre of mass / moments · appears in 13 of 25 sittings, 2016-2025
A composite lamina (or wire framework) is freely suspended from a named point and hangs in equilibrium with a named edge or line at an angle to the downward vertical. Using the centre of mass found earlier in the question, form a moments equation to locate the centre of mass relative to the suspension line, then use trigonometry to relate that position to the angle. The method below applies to the find-angle sittings specifically (the angle is what the student solves for); a further 10 M2 and 1 M3 sitting share the identical suspended-and-in-equilibrium setup but give the angle and ask for a length or mass instead - see familyNote.
The suspended-or-pivoted, hangs-freely-in-equilibrium setup itself (any of the three closing shapes) recurs in 23 of 25 M2 sittings and 4 of 18 M3 sittings - close to a fixture of M2. Of those, 10 M2 sittings and 1 M3 sitting give the angle (or say a line hangs vertical or horizontal) and ask for a length or mass instead (see closingFamily).
The method
7 marks at its longest. Each step below is a mark the examiner is looking for.
State the mass ratio (or vertical distance) of each component part of the composite shape, relative to a common reference (1 mark)
Take moments about the suspension line (or a parallel axis through it); the equation must be dimensionally correct and use every term (1 mark)
Solve the moments equation for the perpendicular distance of the centre of mass from the suspension line (2 marks)
Use trigonometry: the tangent of the angle to the vertical equals the ratio of that perpendicular distance to the other known distance along the suspension line (1 mark)
Solve for the angle to the downward vertical, using the tangent ratio from the previous step (this method's scope is the find-angle sittings; the 7 M2 and 1 M3 sittings that give the angle and ask for a length or mass instead follow the same first four steps but end differently - see familyNote) (2 marks)
What changes between years
Everything below varies from sitting to sitting. The method above does not.
the object: lamina, framework, template, solid (M3's cones/hemispheres/composite solids of revolution, suspended the same way a lamina is), or (2025-January M2) a composite "model" built from a template and two attached laminas
the suspension verb/state: "freely suspended ... hangs in equilibrium" (the majority) vs "hangs freely in equilibrium" (same meaning, reordered - 2016-October, 2017-October, 2018-January, 2018-October) vs "freely pivoted ... hangs in equilibrium" (2023-October) vs "suspended ... hangs freely in equilibrium" (M3 2018-June, of a solid) vs "freely suspended ... hangs at rest" (2025-June)
the suspension point letter and the reference-line letters - always different across sittings
the angle's symbol and form: beta radians, alpha degrees, theta degrees (the majority), an exact tan, or an unnamed exact angle
which quantity is the unknown: 15 of 23 sittings (the find-angle group) give the geometry and ask for the angle (or its tangent) numerically; 8 of 23 invert the question - 6 GIVE a numeric angle (or its tangent) and ask the student to solve for one of the shape's own dimensions or a mass ratio k instead (2016-January, 2016-October, 2018-October, 2022-October, 2023-June M2, plus M3's 2018-June), and 2 GIVE a qualitative condition equivalent to a fixed angle ("AD horizontal", "AC vertical") rather than a numeric one (2017-January, 2017-June M2). Fixed 2026-09-25: an earlier version of this list named 2018-June M2 as inverted; it is not - that sitting's closing step asks for the angle itself (find-angle) and was a transcription slip, not a reclassification. See appearsIn.closingFamily for the exact tally and evidence[].closing for the per-sitting field.
the marks available for this part range from 3 (2023-January, 2023-June) to 10 (2016-January, an unstructured single-part question that folds the whole method - implicit centre of mass, moments equation and trig - into one mark allocation with no lettered parts), shortest when an earlier part already leaves a single clean trig ratio
the closing step: most sittings go straight from the moments-derived distance to tan θ by a single right-triangle ratio; 2025-June M2's part (a) instead supplies the ratio via a separate right-angle geometric relationship between two suspension configurations, so part (b) solves for tan θ through a compound-angle identity (tan(45°-α) manipulation) rather than reading it straight off - a genuinely different final algebraic step attached to the same suspended-object-at-an-angle item model, not a mismatch of the task itself
whether the question is split into lettered parts at all: 2021-January M2's Q4 and 2016-January M2's Q5 pose the entire item - including finding the centre of mass - as a single unlettered question worth 9 and 10 marks respectively, rather than splitting a centre-of-mass part from the suspension-and-angle part
Where the marks go
Instructions the examiners are given, repeated across sittings.
Condone sign errors / condone use of a parallel axis on the moments-equation method mark (6 sittings)
Follow-through (ft) marks on the distance and trig equation: a wrong earlier value does not lose the later marks if carried through correctly (4 sittings)
The final answer must be given in the exact form the question asks for (degrees, radians, or an exact tan) - e.g. 2023-January's mark scheme states 0.82 radians (correct but wrong units) scores M1A1A0, and warns "DO NOT ISW" against a candidate who finds the correct angle then wrongly subtracts it from 90 (1 sittings)
Dependent method marks (DM1) chain the trig step to the success of the earlier moments-equation step (2 sittings)
Find-angle sittings only, matching the M2 headline (2017-June, 2021-January, 2024-January). A fourth M3 sitting (2018-June) has the same setup with the angle given; the M3 family is 4 of 18. GOAL-15 read the two M3 papers not read before (2019-January, 2020-October); neither contains the item.