M1 January 2006 Q7

EdexcelOld spec14 marksConnected Particles Resolving Forces

Figure 3: wedge with two faces at 30 degrees, A (3m) on the smooth face and B (m) on the rough face, string over a pulley at the top — Figure 3

A fixed wedge has two plane faces, each inclined at 30\(^\circ\) to the horizontal. Two particles \(A\) and \(B\), of mass \(3m\) and \(m\) respectively, are attached to the ends of a light inextensible string. Each particle moves on one of the plane faces of the wedge. The string passes over a small smooth light pulley fixed at the top of the wedge. The face on which \(A\) moves is smooth. The face on which \(B\) moves is rough. The coefficient of friction between \(B\) and this face is \(\mu\). Particle \(A\) is held at rest with the string taut. The string lies in the same vertical plane as lines of greatest slope on each plane face of the wedge, as shown in Figure 3.

The particles are released from rest and start to move. Particle \(A\) moves downwards and \(B\) moves upwards. The accelerations of \(A\) and \(B\) each have magnitude \(\tfrac{1}{10}g\).

(a) By considering the motion of \(A\), find, in terms of \(m\) and \(g\), the tension in the string. (3)

(b) By considering the motion of \(B\), find the value of \(\mu\). (8)

Diagram: forces T, R, F and mg on B — Mark scheme (b)

Scheme	Marks
\(A\): \(3mg\sin 30 - T = 3m.\tfrac{1}{10}g\)	M1 A1
\(\Rightarrow T = \tfrac{6}{5}mg\)	A1
	(3)

Scheme	Marks
\(F\): R(perp): \(R = mg\cos 30\)	M1 A1
R(//): \(T - mg\sin 30 - F = m.\tfrac{1}{10}g\)	M1 A2, 1, 0
Using \(F = \mu R\)	M1
\(\dfrac{6}{5}mg - \dfrac{1}{2}mg - \mu mg\dfrac{\sqrt{3}}{2} = \dfrac{1}{10}mg\)	M1
\(\rightarrow \mu = 0.693\) or 0.69 or \(\dfrac{2\sqrt{3}}{5}\)	A1
	(8)

Scheme	Marks
Magn of force on pulley \(= 2T\cos 60 = \tfrac{6}{5}mg\)	M1 A1ft
Direction is vertically downwards	B1 (cso)
	(3)
	(14 marks)