M1 June 2014 (R) Q7

Figure 4: particle P on a plane inclined at 40 degrees, with a 15 N force at 50 degrees to the plane — Figure 4

A particle \(P\) of mass 2.7 kg lies on a rough plane inclined at 40\(^\circ\) to the horizontal. The particle is held in equilibrium by a force of magnitude 15 N acting at an angle of 50\(^\circ\) to the plane, as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the plane. The particle is in equilibrium and is on the point of sliding down the plane.

Find

(a) the magnitude of the normal reaction of the plane on \(P\), (4)

(b) the coefficient of friction between \(P\) and the plane. (5)

The force of magnitude 15 N is removed.

Force diagram: R perpendicular to the plane, F up the plane, mg vertically down, 15 N at 50 degrees to the plane, plane at 40 degrees — Mark scheme (a)

Scheme	Marks
Perpendicular to the slope: \(R = 2.7g\cos 40 + 15\cos 40\)	M1A2
\(=\) 31.8 (N) or 32 (N)	A1
	(4)

Mark scheme (b)

Scheme	Marks
Parallel to the slope: \(F = 2.7g\sin 40 - 15\cos 50\) \((F = 7.366..)\)	M1A2
Use of \(F = \mu R\)	M1
\(\mu = \dfrac{2.7g\sin 40 - 15\cos 50}{R} = 0.23\) or 0.232	A1
	(5)

Notes

N.B. Only penalise over- or under-accuracy after using \(g = 9.8\), (or use of \(g = 9.81\)), once in whole question.

First M1 for resolving parallel to the slope, with correct no. of terms, and both the \(2.7g\) and 15 terms resolved.

First A2 for a correct equation; -1 each error.

Second M1 for use of \(F = \mu R\)

Third A1 for 0.23 or 0.232

Mark scheme (c)

Scheme	Marks
Component of wt parallel to slope \(= 2.7g\sin 40^\circ\ (= 17.0)\)	B1
\(F_{\max} = 0.232 \times 2.7 \times g \times \cos 40^\circ = 4.7\ldots\) (N)	M1A1
\(17.0 > 4.70\) so the particle moves	A1
	(4)
	(13 marks)

Notes

N.B. Only penalise over- or under-accuracy after using \(g = 9.8\), (or use of \(g = 9.81\)), once in whole question.

B1 for component of weight down the plane \(2.7g\sin 40^\circ\) (17 or better)

M1 for using their NEW \(R\) and \(\mu\) to find max friction (M0 if they use \(R\) from (a))

First A1 for 4.7 (or better) (should be 4.701242531)

Second A1 for comparison and correct conclusion.

N.B. If first A mark is 0, the second A mark must also be 0.