Chapter 15 Oscillatory Motion

Intro Periodic Motion- the motion of an object that regularly repeats There is special case of periodic motion in which there is a force acting on an object proportional to the relative position of the object from equilibrium, and that force is directed towards equilibrium.

Intro A force such as this is called a restoring force, and results in Simple Harmonic Motion.

15.1 Motion of an Object Attached to a Spring 1 st Example will be a block of mass m, attached to a spring of elastic constant k. We’ve already studied the qualities of this system in Chapter 7, using Hooke’s Law and the work done by/on the spring.

15.1 In this case, the spring provides the restoring force necessary for simple harmonic motion. If we apply Newton’s 2 nd Law

15.1 By definition, an object moves with SHM whenever its acceleration is proportional to its position and is oppositely directed to the displacement from equilibrium. Note: The SHM motion of a spring also applies to a vertically aligned spring AFTER the spring is allowed to stretch to its new equilibrium. Quick Quiz p. 454

15.2 Mathematical Representation of SHM Remember that acceleration is the 2 nd derivative of position with respect to time. We denote the term k/m with the symbol ω 2 which gives

15.2 We’re thinking integrate this twice to get the position function, but we haven’t seen an integration where a derivative contains the original function. This is called a second order differential equation. To solve this, we need a function x(t) that has a 2 nd derivative that is negative of the original function, multiplied by ω 2.

15.2 The cosine and sine functions do just that. – Derivative of cos = -sin – Derivative of -sin = -cos The full solution to this 2 nd order differential is Where A, ω, and φ are constants representing amplitude, angular frequency and phase angle respectively.

15.2 Amplitude (A) - the maximum distance from equilibrium (in the positive or negative direction). Angular frequency ( ω ) – measures how rapidly oscillations are occurring (rad/s) – For a spring Phase Angle (φ) – uniquely determines the position of the object at t=0.

15.2 The term ( ω t + φ) represents the phase of oscillation. Keep in mind, the cosine function REPEATS every time ω t increases by 2 π radians.

15.2

Quick Quizzes p. 456 Period (T) – the time for one oscillation. Frequency (f) – inverse of period, the number of oscillations per second. (Hz = s -1 )

15.2 Position function Velocity function Acceleration function

15.2 Max values (occur when trig function = 1) – Position- Amplitude – Velocity- (passing through equilibrium) – Acceleration- (at +/- Amp)

15.2 Plots of x(t), v(t), a(t) When x(t) is at + amp – v(t) = 0 – a(t) = -a max When x(t) = 0 – v(t) = +/- v max – a(t) = 0

15.2 Quick Quizzes p. 456 Remember, the angular frequency ( ω ) is determined by (k/m) 1/2 A and φ are determined by conditions at t=0 Examples

15.3 Energy of the Simple Harmonic Oscillator The kinetic energy of the SHO varies with time according to The Elastic Potential Energy of the SHO also varies with time according to

15.3 The total energy is therefore And simplifying… The total mechanical energy of a SHO is a constant of the motion and is proportional to the square of the amplitude.

15.3 Plots of Kinetic and Potential Energy

15.3 From the Energy equations we can determine velocity as a function of position. Solve for v See Figure Pg 463 Example 15.4 pg 464

15.4 Comparing SHM and UCM Simple Harmonic motion can easily be used to drive uniform circular motion. Piston- steam engine (train) internal combustion engine

15.4 Simple Harmonic Motion along a straight line can be represented by the projection of uniform circular motion along the diameter of a reference circle.

15.4 Uniform circular motion is often considered to be a combination of two SHM’s. – One along the x axis. – One along the y axis. – The motions are 90 o or π/2 out of phase. Quick Quiz p 467 Example 15.5

15.5 Pendulums Simple Pendulum exhibits periodic motion, very close to a simple harmonic motion for small angles (<10 o ) The solution to the 2 nd order differential for pendulums is Where

15.5 The period of oscillation is given as The period of a simple pendulum is independent of the mass attached, solely depending on the length and gravity. Quick Quizzes p. 469 Example 15.6

15.5 Physical Pendulum- an object that cannot be approximated as a point mass, oscillating through an axis that does not pass through its center of mass. The moment of inertia of the system must be accounted for

15.5 The period is given as Example 15.7 p 470

15.5 Torsional Pendulum- the oscillation involves rotation, when the support wire twists. The object is returned to equilibrium by a restoring Torque.

15.5 Where κ (kappa) is the torsion constant, often determined by applying a known torque to twist the wire through a measurable angle θ. Using Newton’s 2 nd Law for Torques Rearranged

15.5 We see that this is the same form as our 2 nd Order differential for the mass/spring system, with κ /I representing ω 2. There is no small angle restriction for the torsional pendulum, so long as the elastic limit of the wire is not exceeded.

15.6 Damped Oscillations So far we have discussed oscillations in ideal systems, with no loss of energy. In real life, energy is lost mostly to internal forms, diminishing the total ME with each oscillation. An oscillator whose energy diminishes with time is said to be Damped.

15.6 One common example is when the oscillating object moves through a fluid (air), creating a resistive force as discussed in Ch 6. At low speeds the resistive or retarding force is proportional to the velocity as Where b is called the damping constant.

15.6 Applying Newton’s 2 nd Law Again a 2 nd Order Differential, the solution is a little more complicated. Where

15.6 Recognize that (k/m) 1/2 would be the angular frequency in the absence of damping, called the “Natural Angular Frequency” ( ω o ) The motion occurs within the restricting envelope of the exponential decay.

15.6

Damping falls into three categories based on the value of b. – Underdamped- when the value of bv max < kA As the value of b increases the ampltitude of the oscillations decreases rapidly. – Critically Damped- when b reaches the critical value bc such that When the object is released it will approach but never pass through equilibrium

15.6 – Overdamped- when the value of bv max > kA and The fluid is so viscous that there is no oscillation, the object eventually returns to equilibrium.

15.6 Quick Quiz p. 472

15.7 Forced Oscillations Because most oscillators experience damping, an external force can be used to do positive work on the system compensating for the lossed energy. Often times the value of the force varies with time, but the system will often reach a steady state where the energy gained/lost is equal.

15.7 The system will oscillate according to With a constant Amplitude given by

15.7 Notice when ω≈ω o, the value of A will increase significantly. Essentially when the Forcing frequency matches the natural frequency oscillations intensify. – Examples- Loma Prieta “World Series” Earthquake Tacoma Narrows Bridge