Figure 4 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at 30\(^\circ\) to the horizontal. The string passes over a small smooth light pulley \(A\) fixed at the top of the plane. The part of the string from \(P\) to \(A\) is parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below \(A\). The system is released from rest with the string taut.
(a) Write down an equation of motion for \(P\) and an equation of motion for \(Q\). (4)
(b) Hence show that the acceleration of \(Q\) is 0.98 m s\(^{-2}\). (2)
(c) Find the tension in the string. (2)
(d) State where in your calculations you have used the information that the string is inextensible. (1)
On release, \(Q\) is at a height of 0.8 m above the ground. When \(Q\) reaches the ground, it is brought to rest immediately by the impact with the ground and does not rebound. The initial distance of \(P\) from \(A\) is such that in the subsequent motion \(P\) does not reach \(A\). Find
(e) the speed of \(Q\) as it reaches the ground, (2)
(f) the time between the instant when \(Q\) reaches the ground and the instant when the string becomes taut again. (5)
Mark scheme (a)
Scheme
Marks
N2L \(Q\) \(2g - T = 2a\)
M1 A1
N2L \(P\) \(T - 3g\sin 30^\circ = 3a\)
M1 A1
(4)
Mark scheme (b)
Scheme
Marks
\(2g - 3g\sin 30^\circ = 5a\)
M1
\(a = 0.98\) (ms\(^{-2}\)) * cso
A1
(2)
Mark scheme (c)
Scheme
Marks
\(T = 2(g - a)\) or equivalent
M1
\(\approx 18\) (N) accept 17.6
A1
(2)
Mark scheme (d)
Scheme
Marks
The (magnitudes of the) accelerations of \(P\) and \(Q\) are equal
