M2 January 2005 Q1

EdexcelOld spec7 marksMoments Resolving Forces

Figure 1: horizontal rod AB of length 8a, pivot at A, string from B to C, with C vertically above A and AC = 6a — Figure 1

A uniform rod \(AB\), of length \(8a\) and weight \(W\), is free to rotate in a vertical plane about a smooth pivot at \(A\). One end of a light inextensible string is attached to \(B\). The other end is attached to point \(C\) which is vertically above \(A\), with \(AC = 6a\). The rod is in equilibrium with \(AB\) horizontal, as shown in Figure 1.

(a) By taking moments about \(A\), or otherwise, show that the tension in the string is \(\tfrac{5}{6}W\). (4)

(b) Calculate the magnitude of the horizontal component of the force exerted by the pivot on the rod. (3)

Scheme	Marks
\(M(A)\) \(W \times 4a = T \times 8a\sin\theta\)	M1 A1
Using a value of \(\sin\theta\) and solving	M1
\(T = \tfrac{5}{6}W\ \ *\) cso	A1
	(4)

Scheme	Marks
\(\rightarrow\) \(X = T\cos\theta\)	M1 A1
\(= \tfrac{2}{3}W\)	A1
	(3)
	(7 marks)