Chapter 14 Oscillations Chapter Goal: To understand systems that oscillate with simple harmonic motion. Slide 14-2
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Oscillatory Motion Objects that undergo a repetitive motion back and forth around an equilibrium position are called oscillators. The time to complete one full cycle, or one oscillation, is called the period T. The number of cycles per second is called the frequency f, measured in Hz: 1 Hz = 1 cycle per second = 1 s 1 Slide 14-21
Example 14.1 Frequency and Period of a Loudspeaker Cone Slide 14-22
Simple Harmonic Motion A particular kind of oscillatory motion is simple harmonic motion. In figure (a) an air-track glider is attached to a spring. Figure (b) shows the glider s position measured 20 times every second. The object s maximum displacement from equilibrium is called the amplitude A of the motion. Slide 14-23
Simple Harmonic Motion Figure (a) shows position versus time for an object undergoing simple harmonic motion. Figure (b) shows the velocity versus time graph for the same object. The velocity is zero at the times when x = ± A; these are the turning points of the motion. The maximum speed v max is reached at the times when x = 0. Slide 14-26
Simple Harmonic Motion If the object is released from rest at time t = 0, we can model the motion with the cosine function: Cosine is a sinusoidal function. ω is called the angular frequency, defined as ω = 2π/T. The units of ω are rad/s. ω = 2πf. Slide 14-27
Simple Harmonic Motion The maximum speed is v max = Αω Slide 14-28
Example 14.2 A System in Simple Harmonic Motion Slide 14-29
Example 14.2 A System in Simple Harmonic Motion Slide 14-30
Example 14.2 A System in Simple Harmonic Motion Slide 14-31
Example 14.3 Finding the Time Slide 14-32
Simple Harmonic Motion and Circular Motion Figure (a) shows a shadow movie of a ball made by projecting a light past the ball and onto a screen. As the ball moves in uniform circular motion, the shadow moves with simple harmonic motion. The block on a spring in figure (b) moves with the same motion. Slide 14-33
The Phase Constant What if an object in SHM is not initially at rest at x = A when t = 0? Then we may still use the cosine function, but with a phase constant measured in radians. In this case, the two primary kinematic equations of SHM are: Slide 14-34
The Phase Constant Oscillations described by the phase constants φ 0 = π/3 rad, π/3 rad, and π rad. Slide 14-35
Example 14.4 Using the Initial Conditions Slide 14-36
Example 14.4 Using the Initial Conditions Slide 14-37
Example 14.4 Using the Initial Conditions Slide 14-38
Example 14.4 Using the Initial Conditions Slide 14-39
Energy in Simple Harmonic Motion An object of mass m on a frictionless horizontal surface is attached to one end of a spring of spring constant k. The other end of the spring is attached to a fixed wall. As the object oscillates, the energy is transformed between kinetic energy and potential energy, but the mechanical energy E = K + U doesn t change. Slide 14-46
Energy in Simple Harmonic Motion Energy is conserved in Simple Harmonic Motion: Slide 14-47
Frequency of Simple Harmonic Motion In SHM, when K is maximum, U = 0, and when U is maximum, K = 0. K + U is constant, so K max = U max : So: Earlier, using kinematics, we found that: So: Slide 14-50
Example 14.5 Using Conservation of Energy Slide 14-57
Example 14.5 Using Conservation of Energy Slide 14-58
Example 14.5 Using Conservation of Energy Slide 14-59
Simple Harmonic Motion Motion Diagram The top set of dots is a motion diagram for SHM going to the right. The bottom set of dots is a motion diagram for SHM going to the left. At x = 0, the object s speed is as large as possible, but it is not changing; hence acceleration is zero at x = 0. Slide 14-60
Acceleration in Simple Harmonic Motion Acceleration is the timederivative of the velocity: In SHM, the acceleration is proportional to the negative of the displacement. Slide 14-61
Dynamics of Simple Harmonic Motion Consider a mass m oscillating on a horizontal spring with no friction. The spring force is: Since the spring force is the net force, Newton s second law gives: Since a x = ω 2 x, the angular frequency must be. Slide 14-62
Vertical Oscillations Motion for a mass hanging from a spring is the same as for horizontal SHM, but the equilibrium position is affected. Slide 14-67
Example 14.7 Bungee Oscillations VISUALIZE Slide 14-68
Example 14.7 Bungee Oscillations Slide 14-69
Example 14.7 Bungee Oscillations Slide 14-70
The Simple Pendulum Consider a mass m attached to a string of length L which is free to swing back and forth. If it is displaced from its lowest position by an angle θ, Newton s second law for the tangential component of gravity, parallel to the motion, is: Slide 14-73
The Simple Pendulum If we restrict the pendulum s oscillations to small angles (< 10 ), then we may use the small angle approximation sin θ θ, where θ is measured in radians. and the angular frequency of the motion is found to be: Slide 14-74
Example 14.8 The Maximum Angle of a Pendulum Slide 14-75
Example 14.8 The Maximum Angle of a Pendulum Slide 14-76
Tactics: Identifying and Analyzing Simple Harmonic Motion Slide 14-81
The Physical Pendulum Any solid object that swings back and forth under the influence of gravity can be modeled as a physical pendulum. The gravitational torque for small angles (θ < 10 ) is: Plugging this into Newton s second law for rotational motion, τ = Iα, we find the equation for SHM, with: Slide 14-82
Example 14.10 A Swinging Leg as a Pendulum Slide 14-83
Example 14.10 A Swinging Leg as a Pendulum Slide 14-84
Damped Oscillations An oscillation that runs down and stops is called a damped oscillation. One possible reason for dissipation of energy is the drag force due to air resistance. The forces involved in dissipation are complex, but a simple linear drag model is: The shock absorbers in cars and trucks are heavily damped springs. The vehicle s vertical motion, after hitting a rock or a pothole, is a damped oscillation. Slide 14-87
Damped Oscillations When a mass on a spring experiences the force of the spring as given by Hooke s Law, as well as a linear drag force of magnitude D = bv, the solution is: where the angular frequency is given by: Here is the angular frequency of the undamped oscillator (b = 0). Slide 14-88
Damped Oscillations Position-versus-time graph for a damped oscillator. Slide 14-89
Damped Oscillations A damped oscillator has position x = x max cos(ωt + φ 0 ), where: This slowly changing function x max provides a border to the rapid oscillations, and is called the envelope. The figure shows several oscillation envelopes, corresponding to different values of the damping constant b. Slide 14-90
Mathematical Aside: Exponential Decay Exponential decay occurs in a vast number of physical systems of importance in science and engineering. Mechanical vibrations, electric circuits, and nuclear radioactivity all exhibit exponential decay. The graph shows the function: u = Αe ν/ν 0 = Α exp( ν/ν 0 ) where e = 2.71828 is Euler s number. exp is the exponential function. v 0 is called the decay constant.
Energy in Damped Systems Because of the drag force, the mechanical energy of a damped system is no longer conserved. At any particular time we can compute the mechanical energy from: Where the decay constant of this function is called the time constant τ, defined as: The oscillator s mechanical energy decays exponentially with time constant τ.
Driven Oscillations and Resonance Consider an oscillating system that, when left to itself, oscillates at a natural frequency f 0. Suppose that this system is subjected to a periodic external force of driving frequency f ext. The amplitude of oscillations is generally not very high if f ext differs much from f 0. As f ext gets closer and closer to f 0, the amplitude of the oscillation rises dramatically. A singer or musical instrument can shatter a crystal goblet by matching the goblet s natural oscillation frequency.
Driven Oscillations and Resonance The response curve shows the amplitude of a driven oscillator at frequencies near its natural frequency of 2.0 Hz.
Driven Oscillations and Resonance The figure shows the same oscillator with three different values of the damping constant. The resonance amplitude becomes higher and narrower as the damping constant decreases.
Chapter 14 Summary Slides
General Principles
General Principles
Important Concepts
Important Concepts