Department of Physics United States Naval Academy Lecture 36: Simple Harmonic Motion Learning Objectives • Distinguish simple harmonic motion from other types of periodic motion; analyze SHM by identifying the various rela- tionships between amplitude, period, frequency, phase constant, position, velocity and acceleration. Oscillations: refers to the repetitive or periodic motion of a particle about an equilibrium position. In an oscillating system, the traditional variables x, v, t, and a still apply to motion, but it is more advantageous to introduce new variables that describe the periodic nature of the motion: amplitude, period, and frequency. Amplitude: In an oscillatory motion, a particle or object goes back and forth between two extreme points; the points of maximum displacement from the equilibrium point. Frequency, f of periodic or oscillatory motion is the number of oscillations per second (per unit time). In the SI system, it is measured in hertz: 1 Hz = 1 s−1. Period, T is the time required for one complete oscillation or cycle. It is measured in seconds (s) and is related to the frequency via 1 T = f Simple Harmonic Motion: Is a special kind of oscillatory motion in which the net restoring force acting on an object is proportional to the negative of the displacement from some equilibrium position. A system exhibiting simple harmonic motion is called a Simple Harmonic Oscillator. The displacement, x(t) of a particle from its equilibrium position is described by the equation x(t) = xm cos(wt + f) where xm is the amplitude of the displacement, wt + f is the phase of the motion (this is the argument of the Cosine function and defines the position or location of a point in time), and f is the phase constant. The angular frequency w is related to the period and frequency of the motion by 2p w = = 2p f T The velocity as a function of time of a particle in SHM is obtained by differentiating the displacement equation Figure 1: (a) The displacement x(t) of a particle oscillating in SHM with v(t) = −wxm sin(wt + f) phase angle f equal to zero. The pe- Similarly, the acceleration of a particle in SHM is obtained by differentiating the riod T marks one complete oscillation. velocity equation (b) The velocity v(t) of the particle. (c) a(t) = −w2x cos(wt + f) = −w2 x(t) m The acceleration a(t) of the particle. 2 The quantities, wxm and w xm are the amplitude of the velocity, vm and amplitude of the acceleration, am, respectively; i.e vm = wxm 2 am = w xm © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD The Force Law for Simple Harmonic Motion: The restoring force in simple har- monic motion can be written, according to Newton’s second law, as F = ma = m(−w2x) = −(mw2)x The restoring force always acts in a direction opposite the direction of displace- ment (hence, the negative sign). For the block-spring system, the restoring force is defined by Hooke’s law F = −kx Comparing the two force equations shows that the spring constant can be written as k = mw2 or equivalently, r k w = m w m Figure 2: (Top) A linear simple har- allowing for the calculation of the angular frequency if the mass, and spring’s k monic oscillator on a frictionless sur- constants are known. The period of the motion is then rm face. (bottom) As position changes, T = 2p k the energy shifts between the two types, but the total is constant. Energy in Simple Harmonic Motion: A particle in simple harmonic motion has, 1 2 1 2 at any time, kinetic energy K = 2 mv and potential energy U = 2 kx . If no friction is present, the mechanical energy E = K +U remains constant even though K and U change. For a block-spring system, 1 2 E = K +U = 2 kxm where xm is the amplitude of the spring’s displacement. © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 1.0 A 2.3 kg mass oscillates back and forth from the end of a spring of spring constant 120 N/m. At t = 0, the position of the block is x = 0:13 m and its velocity is vx = −3:4 m/s. (a) What is the angular frequency of the block? (b) What is the mechanical energy of this block-spring system? (c) What is the amplitude of the oscillation? (d) What is the maximum speed of the block and where is this experienced over the motion? (e) What is the phase constant? (Choose a cosine function to describe the motion.) (f) What is the position of block as a function of time? (g) What is the maximum acceleration of the block and where is this experienced over the motion? x = 0:13 m vx = −3:4 m/s © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 2.0 What is the phase constant for the harmonic oscillator with the velocity function v(t) given in the figure below if the posi- tion function x(t) has the form x = xm cos(wt + f)? The vertical axis scale is set by vs = 4:0 cm/s. © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 3.0 A 10 g particle undergoes SHM with an amplitude of 2.0 mm, a maximum acceleration of magnitude 8:0×103 m/s2, and an unknown phase constant f. What are (a) the period of the motion, (b) the maximum speed of the particle, and (c) the total mechanical energy of the oscillator? What is the magnitude of the force on the particle when the particle is at (d) its maximum displacement and (e) half its maximum displacement? © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD Exercise 4.0 If the phase angle for a block-spring system in SHM is p=6 rad and the block’s position is given by x = xm cos(wt + f), what is the ratio of the kinetic energy to the potential energy at time t = 0? © 2018 Akaa Daniel Ayangeakaa, Ph.D., Department of Physics, United States Naval Academy, Annapolis MD.