M1 January 2007 Q6

EdexcelOld spec14 marksForces & Newton's Laws Resolving Forces

Figure 3: box on horizontal ground pulled by a rope at 20 degrees to the ground with tension P N — Figure 3

A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope. The rope makes an angle of 20\(^\circ\) with the ground, as shown in Figure 3. The coefficient of friction between the box and the ground is 0.4. The box is modelled as a particle and the rope as a light, inextensible string. The tension in the rope is \(P\) newtons.

(a) Find the value of \(P\). (8)

The tension in the rope is now increased to 150 N.

(b) Find the acceleration of the box. (6)

Diagram: forces R up, P at 20 degrees above the horizontal, friction mu R and weight 30g — Mark scheme (a)

Scheme	Marks
Use of \(F = \mu R\)	B1
\(\rightarrow\) \(P\cos 20^\circ = \mu R\)	M1 A1
\(\uparrow\) \(R + P\sin 20^\circ = 30g\)	M1 A1
\(P\cos 20^\circ = \mu(30g - P\sin 20^\circ)\)	M1
\(P = \dfrac{0.4 \times 30g}{\cos 20^\circ + 0.4\sin 20^\circ}\)	M1
\(\approx 110\) (N) accept 109	A1
	(8)

Mark scheme (b)

Scheme	Marks
\(\uparrow\) \(R + 150\sin 20^\circ = 30g\) \((R \approx 242.7)\)	M1 A1
N2L \(\rightarrow\) \(150\cos 20^\circ - \mu R = 30a\)	M1 A1
\(a \approx \dfrac{150\cos 20^\circ - 0.4 \times 242.7}{30}\)	M1
\(= 1.5\) (ms\(^{-2}\)) accept 1.46	A1
	(6)
	(14 marks)