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 MECHANICS AND MODELLING MTH-1C32 Time allowed: 2 Hours Attempt QUESTIONS 1 AND 2 and THREE other questions. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTH-1C32 Module Contact: Prof. David Evans, MTH Copyright of the University of East Anglia Version: 2

2 (i) (a) Find the dot product of the following two vectors a = i j, b = 3i+6j k. (b) Find the cross product of the following two vectors c = i+j k, d = i+2j+k. (c) Find the cosine of the angle between the two vectors in part (i). (d) For a general vector u = ui+vj+wk, show that u u = u 2, u u = 0. [13 marks] (ii) (a) The position vector of a particle is r = 5i 3j+7k. Find r i, r j, and r k. (b) By taking an appropriate dot product, find the component of the position vector r in the direction of vector c in part (ii). [7 marks]

3 (i) State fully in words Newton s three laws of motion. [8 marks] (ii) (a) A force F = i + 2j acts on a mass m with position vector r. Write down Newton s second law for the mass. (b) Three forces F 1 = i j + 6k and F 2 = i 3j + k and F 3 = 4j 7k act on a mass m with position vector r. Write down Newton s second law for the mass, and say whether the mass will accelerate or not accelerate. [7 marks] (iii) A mass m is moving at constant velocity ṙ = tj. (a) By integrating the equation ṙ = tj with respect to time, determine the position vector, r(t), given that the mass started at the origin at t = 0. Describe the trajectory of the mass. (b) Using Newton s second law, find the total force acting on the mass. [5 marks] MTH-1C32 PLEASE TURN OVER Version: 2

4 Point particle A travels along the x axis so that it is located at x = X(t) at time t, where X(t) obeys the ordinary differential equation subject to the initial velocity condition d 2 X dt 2 = 1+sint dx dt = 0 at t = 0. The particle starts from the origin so that X = 0 at t = 0. (i) (a) What is the particle acceleration at t = 0? (b) Does the acceleration ever reach zero in t > 0? (c) Sketch the particle acceleration on a graph showing d 2 X/dt 2 against t. (d) Find the velocity of the particle for t > 0. (e) Find the position of the particle for t > 0. [14 marks] (ii) A second particle B is located at x = Y(t), where Y(t) satisfies the differential equation and Y = 1 at t = 0, d 2 Y dt 2 = cost dy dt = 1 at t = 0, (a) In what direction is this second particle moving for small t? (b) Decide if the particles A and B will collide.

5 In an experiment a mass m is fired upwards from the ground at z = 0 at velocity ż = 1. Let g be the acceleration due to gravity acting downwards in the negative z direction. (i) (a) Neglecting air resistance, draw a diagram showing the force acting on the mass during its motion. Include the vertical z axis in your diagram. (b) Show that, according to Newton s second law, the location of the mass z(t) at time t satisfies the ordinary differential equation z = g. (c) Integratethis equationonce tofind ż(t) attime t. Youshould applytheinitial condition that the mass is fired upwards at t = 0 at velocity ż = 1. (d) Integrate again to fnd z(t) and hence determine the maximum height reached by the mass and give the time t M at which this occurs. [14 marks] (ii) A particle of mass m is fired upwards into a viscous fluid at velocity ż = 1 from z = 0. Gravity acts downwards in the negative z direction. The particle also experiences a viscous resistive force of magnitude mµ ż, where µ > 0 is a constant. (a) Find the velocity of the mass, ż, at time t during its motion. (b) Findthetime t V atwhichtheparticlereachesitsmaximumheight. (c) Show that if µ/g is small, then t V t M. MTH-1C32 PLEASE TURN OVER Version: 2

6 (i) (a) Write down Hooke s law for the force F in a spring of stiffness k which is extended an amount x from its natural length. (b) The potential energy in a spring which has been extended by an amount x from its natural length is Calculate V. V = x 0 ksds. (ii) (a) A mass m is suspended in equilibrium from a support by the spring in part (i). Let g be the acceleration due to gravity. Using Hooke s law and equating the force in the spring to the weight of the mass, mg, obtain an expression for e, the equilibrium displacement of the spring. (b) The mass is displaced downwards from equilibrium and released. Let z(t) denote the positive downwards displacement of the mass from the equilibrium position at time t. Using Newton s second law, and making use of the relation for e from part (ii), obtain the following differential equation for z(t), m z +kz = 0. (c) An oscillatory forcing is now included so that m z +kz = cosωt. Find the general solution assuming that ω 2 k/m. (d) Discuss what you expect to happen when ω 2 = k/m. (e) Suppose that ω = k/m+ǫ, where ǫ is small. Show that the amplitude A of the oscillations is given approximately by A 1 2 mk ǫ 1. [14 marks]

7 A simple pendulum consists of a mass m attached to a pivot O by a light rigid rod of length L. Let (r, θ) be polar coordinates with origin at O and with θ measured anticlockwise from zero when the mass is hanging in equilibrium vertically below O. At t = 0 the mass is instantaneously given an angular velocity Ω so that ṙ = 0 and θ = Ω at t = 0. Take r to be the position vector of the mass relative to O. The velocity vector of the mass is given by ṙ = ṙˆr+r θ ˆθ. The acceleration vector of the mass is r = ( r r θ 2 )ˆr+(r θ+2ṙ θ)ˆθ. Let g be the acceleration due to gravity. (i) (a) Write down the velocity vector ṙ of the mass at t = 0. (b) Draw a diagram showing all of the forces acting on the mass at a general time. [6 marks] (ii) (a) Write down Newton s second law for the mass in the transverse and radial directions. (b) Show that the tension in the rod at time t = 0 is given by T = mg +mlω 2. (c) Show that at a general time t, θ + g sinθ = 0. L (d) By first multiplying the equation in (c) by θ, show that the maximum tension in the rod occurs when the mass is directly below O. [4 marks] (e) Show that the mass will at least swing round up to the position where it is vertically above O if g Ω > 2 L. END OF PAPER