Standard +0.3 This is a straightforward centre of mass problem for a composite lamina. Students need to divide the trapezium into standard shapes (typically two triangles or a rectangle and triangle), find individual centres of mass using standard formulas, then apply the composite centre of mass formula. While it requires careful coordinate geometry and arithmetic, it's a routine application of a standard Further Maths technique with no novel insight required.
1 A uniform lamina \(O A B C\) is a trapezium whose vertices can be represented by coordinates in the \(x - y\) plane. The coordinates of the vertices are \(O ( 0,0 ) , A ( 15,0 ) , B ( 9,4 )\) and \(C ( 3,4 )\).
Find the \(x\)-coordinate of the centre of mass of the lamina.
Table of values: Triangle \(OCD\): Area \(= 6\), Distance from \(Oy = 2\); Rectangle \(DEBC\): Area \(= 24\), Distance from \(Oy = 6\); Triangle \(BAE\): Area \(= 12\), Distance from \(Oy = 11\); Trapezium \(OCBA\): Area \(= 42\), Distance from \(Oy = \bar{x}\)
M1
Attempt at moments equation with all necessary terms. Other options possible for RHS of moments equation, for example: (1) \(OAC: 30 \times 6\) and \(ABC: 12 \times 9\); (2) \(OBC: 12 \times 4\) and \(OAB: 30 \times 8\); (3) Subtraction: \(60 \times 7.5 - 6 \times 1 - 12 \times 13\)
1 A uniform lamina $O A B C$ is a trapezium whose vertices can be represented by coordinates in the $x - y$ plane. The coordinates of the vertices are $O ( 0,0 ) , A ( 15,0 ) , B ( 9,4 )$ and $C ( 3,4 )$.
Find the $x$-coordinate of the centre of mass of the lamina.\\
\hfill \mbox{\textit{CAIE Further Paper 3 2022 Q1 [4]}}