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Nov 2014 p43 q6
3941
Particles P and Q have a total mass of 1 kg. The particles are attached to opposite ends of a light inextensible string which passes over a smooth fixed pulley. P is held at rest and Q hangs freely, with both straight parts of the string vertical. Both particles are at a height of h m above the floor (see Fig. 1). P is released from rest and the particles start to move with the string taut. Fig. 2 shows the velocity-time graphs for P's motion and for Q's motion, where the positive direction for velocity is vertically upwards. Find
the magnitude of the acceleration with which the particles start to move and the mass of each of the particles,
the value of h,
the greatest height above the floor reached by particle P.
Solution
(i) From the velocity-time graph, the acceleration can be found using the gradient. The acceleration is 4 m/s². Using Newton's second law for both particles, we have:
\(T - mg = 4m\)
\((1 - m)g - T = 4(1 - m)\)
Solving these equations gives:
\(4 = (1 - m - m)g\)
Thus, P has mass 0.3 kg and Q has mass 0.7 kg.
(ii) Using the area under the velocity-time graph or the formula \(h = \frac{1}{2} a t^2\), we find \(h = 2\) m.
(iii) The distance travelled upwards by P is given by the area under the graph: