Advanced level 2023 south west regional mock pure mathematics with mechanics 3
Advanced level 2023 south west regional mock pure mathematics with mechanics 3
A particle P of muse 5 kg moves with velocity, rms , where v = 3t*i+(1 — 01 at 1 <ccontk CJivcn that ihc
particle is initially at rest at the origin, find.
(a) the speed of P when l – 1 second,
(b) the displacement and acceleration vectors of P at any time f,
(c) the work done in moving the particle from t = 1to t = 3 seconds,
(d) the magnitude of the force acting on P at time t seconds.
-i
Particles P and Q, of masses 0.55 kg and 0.45 kg respectively, arc attached to the ends of a light incxlcnsiblc string which
passes over a smooth. Fixed pulley. The particles arc held at rest with the string taut and its straight parts vertical. Holh
particles arc at a height of 5 m above the ground. The system is released. Find,
(a) the acceleration with which P starts to move and the tension in the string (6 marks)
(2 marks)
The siring breaks al\cr 2 s and in the subsequent motion P and Q move vertically under gravity. At the instant that the string
breaks.Find.
(c) the speed of the particles just when the string breaks.
( d) the height above the ground of P and of Q.
(b) the reaction exerted by the string on the pulley.
The diagram above shows n rectangular lamina OAUC. The coordinates of 0,A, P and C arc (0.0),(8.0),(8,5) and(0,5)
respectively. Particles of mass k m, 5 ?n and 3 m arc attached to the lamina at A,U and C respectively.
The x – coordinate of the centre ofmass of the three particles without the lamina is 6.4
(a) Show that k = 7.
The lamina OAHC is uniform ond has mass 12 mg.
(b) Find the coordinates of the centre of mass of the combined system consisting of the three particles and the lamina
The combined system is freely suspended at O and hangs in equilibrium,
(i ) Find the tangent of the angle which OC makes with the vertical. (2 marks)
Foiccs /•’ “ (i-J)N. h\- (pi+j)N and Fj=(4i+</j)N act at the points with position vectorsr,.r2 and r 3
respectively, where r x = (21- 3J) m,r2 — (-1 4 4)) m andr3 = (41 — 2j) m. Given that the system reduces to a couple.
4 (i )
Find.
(i«) the values of p and 1/.
(hi the magnitude of the moment of the couple.
(3 marks)
(3 marks)
(ii) A paitide of nms m kg hes on a smooth horizontal table and is attached by an inextcnsible string which passes through a
smooth hole in the table, to u particle of mass 2m kg which hangs freely below the table. The particle of mass m kg moves in
a horizontal circle of radius I m on the table with uniform speed such that the panicle of mass 2m kg remains at rest.
Calculate the uniform speed of the particle and the reaction force from ihc table. (7 marks)
