MATHEMATICAL PROBLEM SOLVING, MECHANICS AND MODELLING MTHA4004Y

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1 UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination MATHEMATICAL PROBLEM SOLVING, MECHANICS AND MODELLING MTHA4004Y 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. MTHA4004Y Module Contact: Dr Paul Hammerton, MTH Copyright of the University of East Anglia Version: 1

2 Two vectors a and b are given by a = 2i + 2j k, b = 2i 2k. (i) Calculate a, a b, a b, a (a b). [12 marks] (ii) Find the angle between the two vectors a and b. (iii) The position vector of a particle is given by r = xi + yj + zk. Calculate the component of the r in the direction of vector a. 2. (i) State fully in words Newton s three laws of motion. [8 marks] (ii) Three forces, F 1 = i + 2j k, F 2 = 3i + j + 2k, F 3 = 4i + j k. act on a mass m with position vector r. (a) Using Newton s second law for the mass, calculate the acceleration of the mass. [5 marks] (b) At time t = 0 the mass is located at the origin and has velocity j. If m = 1, calculate the position of the particle at time t = 1 and sketch the path the particle takes for 0 t 1. [7 marks] MTHA4004Y Version: 1

3 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 = 1 + cos t, dt2 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 s acceleration at t = 0? (b) Does the acceleration ever reach zero in t > 0? If so, when? (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 At time t = 0, Y = 1 2 d 2 Y dt 2 + Y = 0. and dy dt = 0. By calculating Y (t), determine whether or not particles A and B collide in t > 0. [6 marks] MTHA4004Y PLEASE TURN OVER Version: 1

4 (i) In an experiment, a mass m is released from rest at the point z = h above the ground at z = 0. Let g be the acceleration due to gravity. You may assume that air resistance can be neglected. (a) Using Newton s second law, show that the location of the mass z(t) at time t satisfies the ordinary differential equation z = g. (b) Integrate once to find ż(t), using the condition that ż(0) = 0. (c) Integrate again to find the vertical position of the mass and hence calculate the time at which the mass hits the ground. (ii) In a second experiment, a mass is dropped from rest at the point x = 0, z = h where x is the coordinate parallel to the ground. In addition to gravity, the mass experiences a force F = mλi mµżk. where λ and µ are positive constants. [12 marks] (a) Find the velocity of the mass v = ẋi + żk. (b) If the mass is released from a sufficiently large height, so h g/µ 2, calculate the speed, v, at which the mass hits the ground. [8 marks] MTHA4004Y Version: 1

5 (i) 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. Give the expression for the potential energy stored in the spring. [5 marks] (ii) A particle of mass m lies on a smooth horizontal table and is constrained to move along the x axis. Initially the mass is located at x = 0 and a spring of natural length l and stiffness k (spring 1) connects the mass to the fixed point x = L where L > l. An identical spring (spring 2) connects the mass to another fixed point x = L. Calculate the force on the mass due to each spring and hence show that the acceleration of the mass is zero. (iii) If the mass is now located at x = X(t), show that the extension of spring 1 is given by x 1 = L l + X and calculate the extension of spring 2. Hence calculate the total force on the mass and show that X(t) satisfies the ordinary differential equation mẍ + 2kX = 0. (iv) The total energy of the system is given by E = T + V where T is the kinetic energy of the mass and V is the total potential energy of the two springs. Show that the total energy is conserved. (v) For a rough table the equation of motion of the mass is given by mẍ + µẋ + 2kX = 0, where µ is a constant satisfying 0 < µ < 8mk. If the mass is released from rest with X(0) = a, sketch X(t) as a function of t, explaining your result. [3 marks] MTHA4004Y PLEASE TURN OVER Version: 1

6 In polar coordinates, a particle of mass m with position vector r, moving with velocity ṙ = ṙ r + r θ θ, satisfies Newton s second law given by F r = m( r r θ 2 ), F θ = m(r θ + 2ṙ θ), where F r and F θ are the forces in the radial and transverse directions respectively. (i) Write down an expression for the kinetic energy of the mass. [3 marks] (ii) A bead of mass m is rolling around the inside of a frictionless bowl which is axisymmetric about the vertical axis. The bead moves on a circular path of radius a in a horizontal plane, with constant angular velocity ω. Gravity acts vertically downwards. Draw carefully a diagram showing all of the forces acting on the bead. (iii) Let φ be the angle between the horizontal and the tangent to the bowl at the point of contact of the mass and R be the normal reaction force between the bead and the bowl. Show that R = mg/ cos φ and hence that ω 2 = g tan φ. [6 marks] a (iv) If the equation of the bowl is given by z = r 2 /c where z is the vertical coordinate and c is a positive constant, show that the time taken for the bead to describe one circuit is 2πλ g, where λ is a constant which you should give. Deduce that the time period is independent of the vertical position of the bead in the bowl. [7 marks] END OF PAPER MTHA4004Y Version: 1