M1 June 2013 (R) Q3

EdexcelOld spec9 marksConnected Particles Resolving Forces

Figure 2: particle A (2 kg) on a plane inclined at 30 degrees, string over pulley P at the top to particle B (4 kg) hanging freely — Figure 2

A fixed rough plane is inclined at 30\(^\circ\) to the horizontal. A small smooth pulley \(P\) is fixed at the top of the plane. Two particles \(A\) and \(B\), of mass 2 kg and 4 kg respectively, are attached to the ends of a light inextensible string which passes over the pulley \(P\). The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs freely below \(P\), as shown in Figure 2. The coefficient of friction between \(A\) and the plane is \(\dfrac{1}{\sqrt{3}}\). Initially \(A\) is held at rest on the plane. The particles are released from rest with the string taut and \(A\) moves up the plane.

Find the tension in the string immediately after the particles are released. (9)

Force diagram: on A, R, T, F and 2g; on B, T up and 4g down; plane at 30 degrees — Mark scheme

Scheme	Marks
Equation of motion of \(B\): \(4g - T = 4a\)	M1A1
Equation of motion of \(A\): \(T - F - 2g\sin 30 = 2a\) OR: \(4g - F - 2g\sin 30 = 6a\)	M1A2
Resolve perpendicular to the plane at \(A\): \(R = 2g\cos 30\)	B1
Use of \(F = \mu R\) : \(F = \dfrac{1}{\sqrt{3}} \times 2g\cos 30 (= g)\)	M1
\(T - g - g = T - 2g = 2a\)
\(2T - 4g = 4g - T\), \(3T = 8g\), \(T = \dfrac{8g}{3} (\approx 26)\) 26.1(N)	DM1A1
	(9)
	(9 marks)

Notes