M2 June 2008 Q6

EdexcelOld spec13 marksCentres of Mass

Figure 3: rectangle OABC with O at the origin, OA = 8 along the x-axis, OC = 5, particles km at A, 5m at B and 3m at C — Figure 3

Figure 3 shows a rectangular lamina \(OABC\). The coordinates of \(O\), \(A\), \(B\) and \(C\) are \((0, 0)\), \((8, 0)\), \((8, 5)\) and \((0, 5)\) respectively. Particles of mass \(km\), \(5m\) and \(3m\) are attached to the lamina at \(A\), \(B\) and \(C\) respectively.

The \(x\)-coordinate of the centre of mass of the three particles without the lamina is 6.4.

(a) Show that \(k = 7\). (4)

The lamina \(OABC\) is uniform and has mass \(12m\).

(b) Find the coordinates of the centre of mass of the combined system consisting of the three particles and the lamina. (6)

The combined system is freely suspended from \(O\) and hangs at rest.

Scheme	Marks
\(M(Oy)\) \((8 + k)m \times 6.4 = 5m \times 8 + km \times 8\)	M1 A1
\(1.6k = 11.2 \ \Rightarrow\ k = 7\) * cso	DM1 A1
	(4)

Mark scheme (b)

Scheme	Marks
\(M(Oy)\) \(27m\bar{x} = 12m \times 4 + 5m \times 8 + 7m \times 8\)	M1 A1
\(\bar{x} = \dfrac{16}{3}\) 5.3 or better	A1
\(M(Ox)\) \(27m\bar{y} = 12m \times 2.5 + 8m \times 5\)	M1 A1
\(\bar{y} = \dfrac{70}{27}\) 2.6 or better	A1
	(6)

Scheme	Marks
\(\tan\theta = \dfrac{\bar{y}}{\bar{x}} = \dfrac{35}{72}\)	M1 A1ft
\(\theta \approx 26^\circ\) awrt \(25.9^\circ\)	A1
	(3)
	(13 marks)