M1 January 2008 Q4

EdexcelOld spec11 marksForces & Newton's Laws Resolving Forces

Figure 1: particle P on a plane inclined at 30 degrees, held by a 49 N force at angle theta to the plane — Figure 1

A particle \(P\) of mass 6 kg lies on the surface of a smooth plane. The plane is inclined at an angle of 30\(^\circ\) to the horizontal. The particle is held in equilibrium by a force of magnitude 49 N, acting at an angle \(\theta\) to the plane, as shown in Figure 1. The force acts in a vertical plane through a line of greatest slope of the plane.

(a) Show that \(\cos\theta = \tfrac{3}{5}\). (3)

(b) Find the normal reaction between \(P\) and the plane. (4)

The direction of the force of magnitude 49 N is now changed. It is now applied horizontally to \(P\) so that \(P\) moves up the plane. The force again acts in a vertical plane through a line of greatest slope of the plane.

Scheme	Marks
R (// plane): \(49\cos\theta = 6g\sin 30\)	M1 A1
\(\Rightarrow\ \ \cos\theta = 3/5\) \(\ast\)	A1
	(3)

Scheme	Marks
R (perp to plane): \(R = 6g\cos 30 + 49\sin\theta\)	M1 A1
\(R \approx 90.1\) or 90 N	DM1 A1
	(4)

Scheme	Marks
R (// to plane): \(49\cos 30 - 6g\sin 30 = 6a\)	M1 A2,1,0
\(\Rightarrow\ \ a \approx 2.17\) or 2.2 m s\(^{-2}\)	A1
	(4)
	(11 marks)