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Simple Harmonic Motion

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Simple Harmonic Motion [SHIVOK SP211] October 30, 2015 CH 15 Simple Harmonic Motion I. Oscillatory motion A. Motion which is periodic in time, that is, motion that repeats itself in time. B. Examples: 1. Power line oscillates when the wind blows past it 2. Earthquake oscillations move buildings C. Sometimes the oscillations are so severe, that the system exhibiting oscillations break apart. 1. Tacoma Narrows Bridge Collapse "Gallopin' Gertie" a) http://www.youtube.com/watch?v=j‐zczJXSxnw II. Simple Harmonic Motion A. http://www.youtube.com/watch?v=__2YND93ofE Watch the video in your spare time. This professor is my teaching Idol. B. In the figure below snapshots of a simple oscillatory system is shown. A particle repeatedly moves back and forth about the point x=0. Page 1 [SHIVOK SP211] October 30, 2015 C. The time taken for one complete oscillation is the period, T. In the time of one T, the system travels from x=+x , to –x , and then back to m m its original position x . m D. The velocity vector arrows are scaled to indicate the magnitude of the speed of the system at different times. At x=±x , the velocity is m zero. E. Frequency of oscillation is the number of oscillations that are completed in each second. 1. The symbol for frequency is f, and the SI unit is the hertz (abbreviated as Hz). 2. It follows that F. Any motion that repeats itself is periodic or harmonic. G. If the motion is a sinusoidal function of time, it is called simple harmonic motion (SHM). 1. Mathematically SHM can be expressed as: 2. Here, a) xm is the amplitude (maximum displacement of the system) b) t is the time c) w is the angular frequency, and d) f is the phase constant or phase angle 3. Figure (a) below plots the displacement of two SHM systems that are different in amplitudes, but have the same period. Page 2 [SHIVOK SP211] October 30, 2015 4. Figure (b) below plots the displacement of two SHM systems which are different in periods but have the same amplitude. a) The value of the phase constant term, , depends on the value of the displacement and the velocity of the system at time t = 0. 5. Figure (c) below plots the displacement of two SHM systems having the same period and amplitude, but different phase constants. 6. For an oscillatory motion with period T, x(t) x(t T ) a) The cosine function also repeats itself when the argument increases by 2. Therefore, b) Here, is the angular frequency, and measures the angle per unit time. Its SI unit is radians/second. To be consistent, then must be in radians. Page 3 [SHIVOK SP211] October 30, 2015 III. Simple Harmonic Motion Graphs A. The displacement equation and graph of SHM: 1. 2. B. The velocity equation and graph of SHM: 1. vt() 2. The maximum value (amplitude) of velocity is x . The phase shift of m the velocity is /2, making the cosine to a sine function. C. The acceleration equation and graph of SHM: 1. at() 2. Page 4 [SHIVOK SP211] October 30, 2015 3. The acceleration amplitude is . 4. In SHM a(t) is proportional to the displacement but opposite in sign. nd IV. Newton’s 2 law for SHM A. SHM is the motion executed by a system subject to a force that is proportional to the displacement of the system but opposite in sign. 1. For example a spring subject to Hooke’s law: 2. 3. The block‐spring system shown above forms a linear SHM oscillator. a) The spring constant of the spring, k, is related to the angular frequency, , of the oscillator: Page 5 [SHIVOK SP211] October 30, 2015 B. Sample Problem for Force law: 1. An oscillator consists of a block of mass 0.500 kg connected to a spring. When set into oscillation with amplitude 35.0 cm, the oscillator repeats its motion every 0.500 s. Find the (a) period, (b) frequency, (c) angular frequency, (d) spring constant, (e) maximum speed, and (f) magnitude of the maximum force on the block from the spring. a) Solution: (a) T = (b) f = (c) = (d) k = (e) vm = (f) Fm = Page 6 [SHIVOK SP211] October 30, 2015 V. Energy in SHM A. The potential energy of a linear oscillator is associated entirely with the spring. B. The kinetic energy of the system is associated entirely with the speed of the block. 1. C. Diagram D. The total mechanical energy of the system: Page 7 [SHIVOK SP211] October 30, 2015 E. Sample Problem: 1. An oscillating block–spring system has a mechanical energy of 1.00 J, an amplitude of 10.0 cm, and a maximum speed of 1.20 m/s. Find (a) the spring constant, (b) the mass of the block, and (c) the frequency of oscillation. a) Solution: (a) (b) (c) Page 8 [SHIVOK SP211] October 30, 2015 I. Pendulums A. In a simple pendulum, a particle of mass m is suspended from one end of an unstretchable massless string of length L that is fixed at the other end. B. First, let’s prove that for a simple pendulum the motion is ALSO simple harmonic motion having the same general solution. Page 9 [SHIVOK SP211] October 30, 2015 C. Second, let’s prove for a simple pendulum the Angular frequency and Period equations. a) This is true for small angular displacements, . b) If we keep < 10 ° we make less than 1 % error. Page 10 [SHIVOK SP211] October 30, 2015 D. Physical pendulum 1. A physical pendulum can have a complicated distribution of mass. If the center of mass, C, is at a distance of h from the pivot point (figure), then for small angular amplitudes, the motion is simple harmonic. I 2. The period, T, is: T 2 mgh a) Here, I is the rotational inertia of the pendulum about O. Page 11 [SHIVOK SP211] October 30, 2015 E. Sample problem: 1. In Fig. below, a stick of length L = 1.85 m oscillates as a physical pendulum. (a) What value of distance x between the stick's center of mass and its pivot point O gives the least period? (b) What is that least period? a) Solution: Page 12 [SHIVOK SP211] October 30, 2015 II. Damped oscillation A. In a damped oscillation, the motion of the oscillator is reduced by an external force. 1. Example: A block of mass m oscillates vertically on a spring on a spring, with spring constant, k. From the block a rod extends to a vane which is submerged in a liquid. The liquid provides the external damping force, F . d 2. Another example, dashpot: st B. Often the damping force, F , is proportional to the 1 power of the d velocity v (remember coffee filter demo in Demo Day #1). That is, Page 13 [SHIVOK SP211] October 30, 2015 nd C. From Newton’s 2 law, the following Diff Eq results: Or D. Let’s think about the solution: 1. We know that the solution to mx kx 0 is the SHM general equation: 2. We know that the dashpot or drag is robbing some energy so there must be a decay of Amplitude over time too, yes? This leads us to the form: bt 2m a) Notice that the Amplitude Xm is multiplied by a factor, xem (). bt This factor is the decay envelope ()e 2m (the dotted lines on the graph below). b) Notice too that the Angular frequency is not the same. It is now Omega Prime. We will discuss this more shortly. 3. Look at the graph of the Cosine function decaying bound by the asymptotes of Page 14 [SHIVOK SP211] October 30, 2015 a) The figure shows the displacement function x(t) for the damped bt 2m oscillator described before. The amplitude decreases as xem () with time. 4. ’ (damped angular frequency) is the new resultant angular frequency, and is given by: bkb2 '()22 = 24mmm2 5. Total Energy decreases with time similar to the amplitude (but with a different multiplier), so: 1 a) Since E UK kX2 2 m 112bt bt b) Then E()tkXe22 (2mm ) kXe ( ) 22mm Page 15 [SHIVOK SP211] October 30, 2015 E. Damped Oscillator sample problem: 1. For a damped simple harmonic oscillator, the block has a mass of 1.2 kg and the spring constant is 5.5 N/m. The damping force is given by – b(dx/dt), where b = 260 g/s. The block is pulled down 10.3 cm and released. (a) Calculate the time required for the amplitude of the resulting oscillations to fall to 1/5 of its initial value. (b) How many oscillations are made by the block in this time? Solution part a: Solution part b: Page 16 [SHIVOK SP211] October 30, 2015 III. Forced oscillations and resonance A. When the oscillator is subjected to an external force that is periodic, the oscillator will exhibit forced/driven oscillations. 1. Example: A swing in motion is pushed with a periodic force of angular frequency, . d B. There are two frequencies involved in a forced driven oscillator: 1. w, the natural angular frequency of the oscillator, without the presence of any external force, and 2. wd, (Drive angular frequency) the angular frequency of the applied external force. C. Resonance will occur in the forced oscillation if the natural angular frequency, , is equal to . d 1. This is the condition when the velocity amplitude is the largest, and to some extent, also when the displacement amplitude is the largest; the below figure plots displacement amplitude as a function of the ratio of the two frequencies.
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