PHYS 211 2014 Exam 1 Your name: ______Solutions__________________________________________ Each problem is worth 10 points. You will only get complete credit for a problem if you fully explain your answer. Problem 1 A particle of mass m slides in a frictionless semicircular depression in the ground of radius R. Find the angular frequency of small amplitude oscillations about the particle's equilibrium position, assuming that the oscillations are essentially one-dimensional, so that the particle passes through the lowest point of the depression during each oscillation cycle. Hint: consider the mechanical energy of the mass. The motion of the mass is essentially the same as that of a simple pendulum. Let θ be the angular displacement of the mass from its equilibrium position. The mechanical energy of the mass is E= 1 mR 2θɺ 2 + mgR (1 − cos θ ) , 2 where the zero level for gravitational potential energy has been taken to be in the equilibrium position of the mass. i.e. at the bottom of the depression. For small oscillations, the cosine can be replaced by the first two term of its series expansion, which gives E= 1 1 mR 2θɺ 2 + mgRθ 2 . 2 2 By making the substitution x = Rθ , the energy becomes the same form for the SHO, 1 2 1 g 2 mxɺ + m x . 2 2 R = mg R , and so the frequency of small oscillations is E= The effective spring constant is keff ω= keff = m g . R Alternatively, we can use conservation of mechanical to derive the equation of motion, dE ɺ = θ mR 2θɺɺ + mgRθ = 0, dt ( ) which gives g R θɺɺ + θ = 0. Problem 2 A particle executes two-dimensional simple harmonic oscillation such that its instantaneous coordinates in the x-y plane are x ( t ) = a cos (ωt ) , y ( t ) = a cos (ωt − φ ) . Describe the motion in the x-y plane when (a) φ = 0, (b) φ = π/4, and (c) φ = π/2. In each case, sketch the trajectory of the particle in the x-y plane. (a) When φ = 0, we have y ( t ) = x ( t ) , which means that the 1.0 motion in the x-y plane is a straight line segment as shown at right. The trajectory is shown for a =1. y 0.5 0.0 -0.5 -1.0 -1.0 -0.5 0.0 x 0.5 1.0 (b) When φ = π/4, we find, by using a graphing calculator, that the trajectory in the x-y plane is an ellipse, with axes rotated by π 4 radians in the x-y plane. The trajectory is shown for a =1. 1.0 y 0.5 0.0 -0.5 -1.0 -1.0 -0.5 0.0 0.5 1.0 x (c) When φ = π/2, we have 1.0 x ( t ) = a cos (ωt ) , y ( t ) = a sin (ωt ) . 2 2 0.5 2 Hence x + y = a . The trajectory is a shown for a =1. y circle of radius a. The trajectory is 0.0 -0.5 -1.0 1.0 0.5 0.0 x -0.5 -1.0 Problem 3 Two masses are (separately) suspended from a rigid support by identical springs. The masses are of equal shape and size, but m2 is twice as heavy as m1. You may assume that the damping forces on the masses are only determined by their shape and size, not by their mass. We observe that the amplitude of oscillation of the first (lighter) mass goes down by a factor of 4 over a time of 35 seconds. How long does it take for the amplitude of the second mass to be reduced by the same factor? − b t The amplitude of the oscillation decays according to A = A0 e 2 m , where A0 is the initial amplitude and b is the damping constant. Both masses have the same value for b but differ in mass. For the lighter mass, we are given that b − τ1 1 = e 2 m1 , 4 where τ 1 = 35 s. The time, τ 2 , for the amplitude of the heavier mass to decay by the same factor is given by b − τ2 1 = e 2 m2 . 4 Hence τ 2 m2 = τ 1 m1 . This gives the time for the amplitude of the second mass to be reduced by the same factor as the first mass to be 70 s.