M1 June 2005 Q3

Figure 1: string from A to B and from B up to C, with A and C level, angle BAC = alpha and a horizontal force of 6 N at B — Figure 1

A smooth bead \(B\) is threaded on a light inextensible string. The ends of the string are attached to two fixed points \(A\) and \(C\) on the same horizontal level. The bead is held in equilibrium by a horizontal force of magnitude 6 N acting parallel to \(AC\). The bead \(B\) is vertically below \(C\) and \(\angle BAC = \alpha\), as shown in Figure 1. Given that \(\tan\alpha = \tfrac{3}{4}\), find

(a) the tension in the string, (3)

(b) the weight of the bead. (4)

Scheme	Marks
R(\(\rightarrow\)) \(T\cos\alpha = 6\)	M1 A1
\(\rightarrow T = 7.5\) N	A1
	(3)

Scheme	Marks
R(\(\uparrow\)) \(T + T\sin\alpha = W\)	M1 A1
Using same \(T\)’s and solving	M1
\(\rightarrow W = 12\) N	A1
	(4)
	(7 marks)