M1 June 2010 Q7

Figure 2: particle on a plane inclined at angle alpha, held by a horizontal force P N — Figure 2

A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude \(P\) newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \dfrac{3}{4}\), as shown in Figure 2.

The coefficient of friction between the particle and the plane is \(\dfrac{1}{3}\).

Given that the particle is on the point of sliding up the plane, find

(a) the magnitude of the normal reaction between the particle and the plane, (5)

(b) the value of \(P\). (5)

Scheme	Marks
\(F = \tfrac{1}{3}R\)	B1
\((\uparrow)\ \ R\cos\alpha - F\sin\alpha = 0.4g\)	M1 A1
\(R = \tfrac{2}{3}g = 6.53\) or 6.5	M1 A1
	(5)

Scheme	Marks
\((\rightarrow)\ \ P - F\cos\alpha - R\sin\alpha = 0\)	M1 A2
\(P = \tfrac{26}{45}g = 5.66\) or 5.7	M1 A1
	(5)
	(10 marks)