A-LevelMathematicsMechanics9709/52 · Oct/Nov 2018 · Q67 Marks

The diagram shows a uniform lamina ABCDEFGH. The lamina consists of a quarter-circle OAB of radius r m, a rectangle DEFG and two isosceles right-angled triangles COD and GOH. The rectangle has DG = EF = r m and DE = FG = x m. [Figure 6.1]

(i)Given that the centre of mass of the lamina is at O, express x in terms of r.[6]
(ii)Given instead that the rectangle DEFG is a square with edges of length r m, state with a reason whether the centre of mass of the lamina lies within the square or the quarter-circle.[1]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

[B1] Centre of mass of triangles below O = r/6
[B1] Centre of mass of quadrant below O = (2rsinπ/4)/(3π/4)
[M1] Attempt to take moments about O
[A1] (rx)(x/2) = (r²/4)(r/6)+ (πr²/4)(2rsinπ/4)/(3π/4)
[M1] Attempt to express x in terms of r
[A1] x = 1.01r
[B1ft] Within quadrant as the square will be smaller than the rectangle Or if x < r in part (i), within the square as the square will be larger than the rectangle

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