Do not turn over until you are told to do so by the Invigilator.

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1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination ENGINEERING MATHEMATICS AND MECHANICS ENG-4004Y Time allowed: 2 Hours Attempt QUESTIONS 1 and 2, and ONE other question. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Inviilator. ENG-4004Y Module Contact: Dr Rob Penfold, MTH Copyriht of the University of East Anlia Version: 1

2 A particle A travels alon the x -axis so that it is located at x a (t at time t 0, where x a (t obeys the ordinary differential equation ẍ a (t = 48t, subject to the initial velocity condition ẋ a (0 = 4. oriin so that x a = 0 at t = 0. The particle starts from the (i (a Sketch the particle acceleration on a raph showin ẍ a (t aainst t for t 0. (b By interatin the acceleration, find the velocity of the particle for t 0. (c Sketch the particle velocity on a raph showin ẋ a (t aainst t for t 0. (d Does the velocity ever reach zero at a time t > 0? If so, find the positive time when the particle is instantaneously stationary. (e Show that the position x a (t of the particle for t 0 is iven by x a (t = 4t ( 2 t 1 ( 2 t + 1. (ii A second particle B is located at x b (t. Initially at rest with x b (0 = 0, particle B moves with constant acceleration ẍ b = 8. (a Find x b as a function t. (b Determine the instant t > 0 when the particles A and B collide. [7 marks] (iii The motion of a third particle C is iven by ẍ c (t = 48t with ẋ c (0 = 4, x c (0 = 0. (a Without solvin this differential equation, briefly explain how the motion of C is related to the motion of A. (b Determine the instant t > 0 when the particles C and B collide.

3 (i A particle of mass m, with position vector r(t at time t, is acted upon by three constant forces, F 1 = i 2 j 3F k, F 2 = 2 i + 3 j F k, F 3 = 3 i j + 2F k, where F > 0 is a scalar constant. (a By usin Newton s second law of motion 3 m r = F = F n, calculate the acceleration r of the particle in terms of m and the resultant force F. (b At time t = 0 the particle is positioned at the oriin r(0 = 0 with velocity ṙ = uk where u > 0 is a scalar constant. Show that the particle velocity is iven by ( ṙ = u 2F t k, m and hence find an expression for the position r(t of the particle as a function of time t. (c By calculatin the distance r(t, show that the particle has returned to the oriin at time t = mu/f > 0. n=1 (ii A particle is in motion and has position vector r(t = cos ( θ(t i + sin ( θ(t j, where the anle θ(t = 2πt/(1 + t is a function of time t 0. (a Calculate the derivative θ = dθ dt and show that θ(t > 0 for all t 0. (b Verify that r = 1 for all time. (c Find the velocity ṙ of the particle and calculate r ṙ. (d By interpretin the physical context of each result (a, (b and (c, describe the motion of the particle. ENG-4004Y PLEASE TURN OVER Version: 1

4 A model particle of mass m lies on a frictionless horizontal surface. It is attached to two model sprins: the left sprin has stiffness k and natural lenth 2l 0 ; the riht sprin has stiffness 3k and natural lenth l 0. The other ends of the sprins are attached to fixed points alined horizontally, a distance 5l 0 apart. The particle can move alon the line of the sprins. 5l 0 m k, 2l 0 3k, l 0 x (i Write down Hooke s law for the force H in a model sprin of stiffness k and natural lenth l 0. Carefully define any additional symbols that you use. (ii By usin Newton s second law of motion, show that the distance x = x(t of the particle from the left-hand end at time t satisfies the differential equation mẍ + 4kx = 14kl 0. [8 marks] (iii Hence find the distance from the left-hand end at which the particle could remain in equilibrium. (iv By comparin with the standard equation of simple harmonic motion, ẍ + ω 2( x x eq = 0, around equilibrium position x eq the particle s oscillations. with anular frequency ω, find the period T of (v Suppose that the particle is initially (at t = 0 released from rest halfway between the two ends. Find an expression for x(t. (vi Does the model predict that the particle will hit one of the ends in its subsequent motion? Justify your answer.

5 On flat horizontal round, define a coordinate system so that i is directed parallel to the round and j is directed vertically upwards. The oriin is set at round level. At time t = 0 a particle is launched with speed U > 0 from a position r(0 = h j (with h > 0 and directed at an anle α above the horizontal (0 < α < π/2. Assume that ravity, directed downwards, is the only force actin on the particle. Let be the constant manitude of the ravitational acceleration. (i Draw a diaram of the situation and sketch an approximate trajectory of the particle. (ii Write down the initial condition on the particle velocity ṙ(0. (iii Show that, at time t 0, the particle position is iven by r(t = U cos(αt i + ( h + U sin(αt 1 2 t2 j. [4 marks] (iv Show that the fliht time of the particle, from launch to first hittin the round, is T = 1 ( U sin(α + U 2 sin 2 (α + 2h. [4 marks] (v Show that the distance away from the oriin when the particle hits the round is R = U cos(α ( U sin(α + U 2 sin 2 (α + 2h. (vi For a iven U with h = 0, show that the maximum value of R that can be attained by varyin the launch anle α is R max = U 2, where α = π/4. [6 marks] END OF PAPER