M1 June 2016 Q8

EdexcelOld spec12 marksConnected Particles Forces & Newton's Laws

Figure 3: particle P (1.5 kg) on a horizontal table joined by a string over a pulley at the edge to particle Q (3 kg) hanging vertically — Figure 3

Two particles \(P\) and \(Q\) have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough horizontal table. The coefficient of friction between \(P\) and the table is \(\dfrac{1}{5}\). The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle \(Q\) hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle \(P\) is released from rest with the string taut and slides along the table.

Assuming that \(P\) has not reached the pulley, find

(a) the tension in the string during the motion, (8)

(b) the magnitude and direction of the resultant force exerted on the pulley by the string. (4)

Mark scheme (a)

Scheme	Marks
\(F = \tfrac{1}{5}R\)	M1
\(R = 1.5g\)	B1
\(T - F = 1.5a\)	M1 A1
\(3g - T = 3a\)	M1 A1
\(T = 1.2g\) or 11.8 N or 12 N	DM1 A1
	(8)

Notes

First M1 for use of \(F = \tfrac{1}{5}R\) in an equation.

B1 for \(R = 1.5g\)

Second M1 for resolving horizontally with usual rules

First A1 for a correct equation

Third M1 for resolving vertically with usual rules

Second A1 for a correct equation

N.B. Either of the above could be replaced by a whole system equation: \(3g - F = 4.5a\)

N.B. All of the marks for the two equations can be scored if they consistently use \(-a\) instead of \(a\).

Fourth M1 dependent on first, second and third M marks for solving their equations for \(T\)

Third A1 for \(1.2g\), 11.8 (N) or 12 (N)

Mark scheme (b)

Scheme	Marks
\(R = \sqrt{T^2 + T^2}\) or \(2T\cos 45^\circ\) or \(\dfrac{T}{\cos 45^\circ}\)	M1 A1
\(= 16.6\) (N) or 17(N) or \(\dfrac{6g\sqrt{2}}{5}\)	A1
Direction is 45\(^\circ\) below the horizontal oe	B1
	(4)
	(12 marks)

Notes

First M1 for a complete method for finding the magnitude of the resultant (N.B. M0 if different tensions used),

First A1 for \(\sqrt{T^2 + T^2}\) or \(2T\cos 45^\circ\)

Second A1 for 16.6(N) or 17 (N)

B1 for 45\(^\circ\) below the horizontal or a diagram with an arrow and a correct angle. Ignore subsequent wrong answers e.g. a bearing of 225\(^\circ\), which scores B0, as does SW etc.