MATHEMATICAL MODELLING, MECHANICS AND MOD- ELLING MTHA4004Y

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1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination MATHEMATICAL MODELLING, MECHANICS AND MOD- ELLING MTHA4004Y Time allowed: 2 Hours Attempt QUESTIONS 1 and 2, and ONE other question. You will not be penalised if you attempt additional questions. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. MTHA4004Y Module Contact: Professor David Stevens, MTH Copyright of the University of East Anglia Version: 1

2 A particle travels along the x -axis so that it is located at x(t) for time t 0. At t = 0 the particle starts at rest from x = 0 and has acceleration ẍ(t) = t 2 3t + 2 for t 0. (i) What is the particle acceleration at t = 0? (ii) For what values of t > 0 does the acceleration equal zero? (iii) Find the speed of the particle, ẋ(t), for t > 0. (iv) Which direction is the particle moving for small t? (v) Which direction is the particle moving for large t? (vi) Find the position of the particle, x(t), for t > 0. (vii) Does the particle ever return to x = 0? 2. Let x measure horizontal distance along the ground and y the distance vertically above it, with the ground at y = 0. A mass m is projected into the air from the point (x, y) = (0, 0) at a speed U and an angle α to the horizontal. (i) Assuming no air resistance, write the x and y components of Newton s second law for the mass. [5 marks] (ii) Show that the equation for the trajectory of the mass is given by y = g 2U 2 x2 (1 + tan 2 α) + x tan α, where g is the acceleration due to gravity. [10 marks] (iii) Show that the maximum horizontal distance travelled is U 2 g sin 2α. [5 marks] MTHA4004Y Version: 1

3 A particle of mass m lies between two vertical surfaces. It is attached to two springs: the left spring has stiffness k and natural length 2l 0 ; the right spring has stiffness 2k and natural length l 0. The other ends of the springs are attached to fixed points aligned horizontally, a distance 5l 0 apart. The particle can move along the line of the springs. Gravity is neglected. 5l 0 m k, 2l 0 2k, l 0 x (i) Write down Hooke s law for the force H in a spring of stiffness k and natural length l 0. (ii) By using 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 + 3kx = 10kl 0. (iii) Hence find the distance from the left-hand end at which the particle could remain in equilibrium. [3 marks] (iv) By comparing 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 angular frequency ω, find the period T of [3 marks] (v) Suppose that at t = 0 the particle is set in motion from the equilibrium position with initial velocity ẋ(0) = ωl 0. Find an expression for x(t). [6 marks] (vi) Will the particle hit one of the vertical surfaces in its subsequent motion? Justify your answer. MTHA4004Y PLEASE TURN OVER Version: 1

4 A satellite of mass m orbits a planet under the gravitational force mµ/r 2 directed toward the centre O of the planet, where µ is a constant and r is the distance from O. At time t = 0 the satellite is at a point S, a distance a from O, and its velocity has magnitude U directed perpendicular to OS so that, at t = 0, ṙ = 0 and a θ = U. Here θ is the angle between the position vector of the satellite and OS. In polar coordinates, Newton s 2nd law of motion for the satellite is given as (i) Using these equations, show that m( r r θ 2 ) = mµ r 2, m(2ṙ θ + r θ) = 0. r 2 1 θ = h, 2ṙ2 + h2 2r µ 2 r = E, where h and E are undetermined constants. [8 marks] (ii) Using the initial condition, determine h and hence show that E = 1 2 U 2 µ a. (iii) Write ṙ 2 as a function of r. Given that ṙ = 0 when the satellite is at its greatest and least distances from O, deduce that when U 2 = µ/a the satellite must be moving in a circle of radius a and centre O. [8 marks] END OF PAPER MTHA4004Y Version: 1

5 MTHA4004Y Mechanics Examination Feedback 2018 On average, students performed extremely well on compulsory question 1, very well on compulsory question 2, reasonably on optional question 3. Optional question 4 proved unpopular, selected by fewer than 10% of students. Q1. This compulsory question on 1-dimensional kinematics was very straightforward and the majority of students provided excellent solutions. The key steps were integrating the acceleration to find the speed and integrating the speed to find the position. Most of the (few) mistakes were small slips in algebra. Q2. This compulsory question concerned projectile motion with gravity being the only force acting on the projectile. The question closely followed a section of the lecture notes (4.2 in 2017/8). Again most students performed very well. The most common mistake was using the initial velocity as the force when applying Newton s second law of motion. Q3. This optional question concerned the motion of a mass connected to two springs. Most students selected this question and provided reasonable solutions. Part (v) provided the greatest challenge. This required the solution of an inhomogeneous 2 nd order Ordinary Differential Equation (ODE). The correct technique is to find and combine the Complimentary Function (the solution of the associated homogeneous problem) and the Particular Integral (any solution of the full ODE which included the previously determined equilibrium solution). Q4. This optional question concerned the orbit of a satellite. Very few students selected the question. Part (i) concerned conservation of angular momentum, which had been covered in the lecture notes. Part (ii) simply required substitution of the initial conditions into the (provided) results from part (i). A number of students spotted this without completing part (i). The final part examined the specific case of a circular orbit and simply required algebraic manipulation and the solution of a quadratic equation.