OCR Tuesday 19 October 2021 – Afternoon A Level Further Mathematics A Y543/01 Mechanics

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Answer all the questions. 1 One end of a light elastic string of natural length 0.6m and modulus of elasticity 24N is attached to a fixed point O. The other end is attached to a particle P of mass 0 ... .4kg. O is a vertical distance of 1m below a horizontal ceiling. P is held at a point 1.5m vertically below O and released from rest (see diagram). O 1.5m 1m P Assuming that there is no obstruction to the motion of P as it passes O, find the speed of P when it first hits the ceiling. [5] 2 A particle P of mass 2kg is moving on a large smooth horizontal plane when it collides with a fixed smooth vertical wall. Before the collision its velocity is ( ) 5 1 i j + 6 m s-1 and after the collision its velocity is ( ) - + 3i j m s-1. (a) The impulse imparted on P by the wall is denoted by INs. Find the following. • The magnitude of I • The angle between I and i [6] (b) Find the loss of kinetic energy of P as a result of the collision. [3] 3 A particle P of mass m moves on the x-axis under the action of a force F directed along the axis. When the displacement of P from the origin is x its velocity is v. (a) By using the fact that the dimensions of the derivative dd vx are the same as those of vx , verify that the equation dd F mv vx = is dimensionally consistent. [2] It is given that v k = - m a - 21 2 2 x where a and k are constants. (b) Explain why [a] must be the same as [x]. [1] (c) Deduce the dimensions of k. [2] (d) Find an expression for F in terms of x and k. [3]3 © OCR 2021 Y543/01 Oct21 Turn over 4 A hollow cone is fixed with its axis vertical and its vertex downwards. A small sphere P of mass mkg is moving in a horizontal circle on the inner surface of the cone. An identical sphere Q rests in equilibrium inside the cone (see diagram). P Q The following modelling assumptions are made. • P and Q are modelled as particles. • The cone is modelled as smooth. • There is no air resistance. (a) Assuming that P moves with a constant speed vms-1, show that the total mechanical energy of P is mv J 32 2 more than the total mechanical energy of Q. [6 [Show More]

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