M1 June 2011 Q3

Figure 1: particle on a plane inclined at alpha, held by a horizontal force of 4 N — Figure 1

A particle of weight \(W\) newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \dfrac{3}{4}\), as shown in Figure 1.

The coefficient of friction between the particle and the plane is \(\dfrac{1}{2}\).

Given that the particle is on the point of sliding down the plane,

(i) show that the magnitude of the normal reaction between the particle and the plane is 20 N,

(ii) find the value of \(W\). (9)

Mark scheme

Scheme	Marks
\(\nearrow\) \(4\cos\alpha + F = W\sin\alpha\)	M1 A1
\(\nwarrow\) \(R = 4\sin\alpha + W\cos\alpha\)	M1 A1
\(F = 0.5R\)	B1
\(\cos\alpha = 0.8\) or \(\sin\alpha = 0.6\)	B1
\(R = 20\) N ** GIVEN ANSWER	M1 A1
\(W = 22\) N	A1
	(9)
	(9 marks)

OR

\(\rightarrow\) \(R\sin\alpha = 4 + F\cos\alpha\)	M1 A1
\(\uparrow\) \(R\cos\alpha + F\sin\alpha = W\)	M1 A1
\(F = 0.5R\)	B1
\(\cos\alpha = 0.8\) or \(\sin\alpha = 0.6\)	B1
\(R = 20\) N ** GIVEN ANSWER	M1 A1
\(W = 22\) N	A1