M2 June 2006 Q4

EdexcelOld spec10 marksCentres of Mass

Figure 1: rectangle ABCD with AB = 2a and AD = 3a, particles 6m at C and 2m at D — Figure 1

Figure 1 shows four uniform rods joined to form a rigid rectangular framework \(ABCD\), where \(AB = CD = 2a\), and \(BC = AD = 3a\). Each rod has mass \(m\). Particles, of mass \(6m\) and \(2m\), are attached to the framework at points \(C\) and \(D\) respectively.

(a) Find the distance of the centre of mass of the loaded framework from

(i) \(AB\),

(ii) \(AD\).

(7)

The loaded framework is freely suspended from \(B\) and hangs in equilibrium.

(b) Find the angle which \(BC\) makes with the vertical. (3)

Mark scheme (a)

Scheme	Marks
Total mass \(= 12m\) (used)	M1
(i) M(\(AB\)): \(m.3a/2 + m.3a/2 + m.3a + 6m.3a + 2m.3a = 12m.x\) indep	M1 A1
\(\Rightarrow x = \dfrac{5}{2}a\)	A1
(ii) M(\(AD\)): \(m.a + m.a + m.2a + 6m.2a = 12m.y\) indep	M1 A1
\(\Rightarrow y = \dfrac{4}{3}a\)	A1
	(7)

Scheme	Marks
\(\tan\alpha = \dfrac{2a - 4a/3}{5a/2}\)	M1 A1 f.t.
\(\Rightarrow \alpha \approx \underline{14.9^\circ}\)	A1 cao
	(3)
	(10 marks)