M1 January 2013 Q3

3. A particle \(P\) of mass 2 kg is attached to one end of a light string, the other end of which is attached to a fixed point \(O\). The particle is held in equilibrium, with \(OP\) at 30\(^\circ\) to the downward vertical, by a force of magnitude \(F\) newtons. The force acts in the same vertical plane as the string and acts at an angle of 30\(^\circ\) to the horizontal, as shown in Figure 3.

Figure 3: particle P on a string from O at 30 degrees to the downward vertical, force F N at 30 degrees above the horizontal — Figure 3

Find

(i) the value of \(F\),

(ii) the tension in the string. (8)

Mark scheme

Scheme	Marks
\((\uparrow)\), \(T\cos 30 + F\cos 60 = 2g\)	M1 A1
\((\rightarrow)\), \(T\cos 60 - F\cos 30 = 0\)	M1 A1
\(F = g = 9.8\)	M1 A1
\(T = \sqrt{3}g = 17\) or 17.0	M1 A1
	(8 marks)

OR

\((\nearrow)\), \(F = 2g\cos 60\)	M1 A1
\((\nwarrow)\), \(T = 2g\cos 30\)	M1 A1
\(F = g = 9.8\)	M1 A1
\(T = \sqrt{3}g = 17\) or 17.0	M1 A1
	(8 marks)