Chapter 16Oscillations

OscillationsMotion that repeats itself at regular intervals:

SpringPendulumRocking horseDiatomic moleculeKid on a trampolineSpecial case: Simple harmonic motion (SHM)

Simple Harmonic MotionMx = 0Classic case of SHM: Mass on a massless spring with no friction Defines equilibrium length

Simple Harmonic Motion Definedx(t) = xmcos(wt + f)xm = amplitude of oscillationw = angular frequency (radians/second)f = phase of motion (radians) (tells you position and velocity at t = 0)f = w/2p = frequency # of cycles/secondT = 1/f = period xtxm -xm

What is Phase? x(t) = xmcos(t + )xt determines position and velocity at t = 0 (and shifts oscillation at any later time) f = 0 f < 0 f > 0

Visualizing SHM Equivalent to the 1-dimensional projection of an object executing uniform circular motionhttp://www.phy.ntnu.edu.tw/java/shm/shm.html

SHM: Acceleration and Velocity

What Sort of Force Gives SHM?Hookes law!a(t) = -w2x(t)Ftot = ma = -mw2 xForce is proportional to displacement with a negative constant of proportionalityF = -kxmw = (k/m)w is the frequency of oscillation of the massw does not depend on amplitude of motion

Example: Hanging MassWhat about a hanging spring? Does it still obey SHMFtot = mg-kx = maEquilibrium when:mg = kx0x0 = mg/k

Example: Hanging MassWhat is the frequency of oscillation?Same as before withx shifted by x0 = mg/kNeed a solution to:Try:

Energy in a SpringTotal energy is constant, but sloshes between kinetic and potential Kinetic energy: Potential energy:

Energy in a SpringTwo special situations where calculating total energy is easy:All potential energyAll kinetic energy

Simple Pendulumh = L(1-cos)LmFor small angles,

Simple Pendulum Follows SHMLooks like a springSolution by analogySpringPendulum

Simple Pendulum: QuestionsQ1. If we double m, the period:a) is half as large d) is 4 times greaterb) is twice as large e) stays the same c) is 2 times greaterQ2. If we double L, the period:a) is half as large d) is 4 times greaterb) is twice as large e) stays the same c) is 2 times greater

Physical PendulummgCOMdAn object with physical extent:We know the solution from before:Any system with a minimum in energy looks like a SHO near equilibrium

Damped OscillationsSHM is an idealizationEnergy is constantMotion never decaysIn real life the motion eventually stopsEnergy 0Fd = -bvDirection opposite to motionMagnitude proportional to velocityNeed to add a damping force in the equation of motion:

Damped OscillationsSHM equation of motion (no damping)How do we solve this?Adding the damping term:

In real life the system loses energyThe Ideal Case:What Happens in Real Life?Rate of energy loss proportional to energy

The SolutionGuess a solution of the form:Back to the damped oscillatorIt works if:You can fill in the gaps after you learn differential equations!

Damped OscillatorThere are three types of solutionsNo oscillations!UnderdampedOverdampedCritically damped

Underdamped

Overdamped

Critical Damping

Forced OscillationsTwo ways to apply a forceStatic a constant pushDynamic periodic pushConsider a Dynamic Force

The Effects of a ForceExamine limiting casesTf >> TSlowly varying forceFast OscillationsSlow ForceF ~ constant during each oscillationA constant force just shifts the equilibrium position

Rapidly Varying ForcesTf

ResonanceBig response when d= !The equation of motion isThe exact solution is

ResonanceResonance can lead to spectacular consequences!