Lecture 01

Physics 232 Lecture 01 1

Periodic Motion or

Oscillations


Periodic Motion Periodic Motion is motion that repeats about a point of stable equilibrium

A necessary requirement for periodic motion is a Restoring Force

Stable Equilibrium

Unstable Equilibrium


Characteristics

Amplitude - A The maximum displacement from the equilibrium position

Period - T The time to complete one cycle of motion, peak to peak or valley to valley

Frequency - f The number of cycles per unit time

Tf 1=

A

A

T


Simple Harmonic Motion Consider a mass m attached to a horizontal spring having a spring constant k with the spring unstretched

We now pull the mass to the right and then let the mass go

What is the subsequent motion of the mass?

The restoring force is given by Hookes Law xkmaF ==


Simple Harmonic Motion This is a second order differential equation

xkdt

xdm =22

Since this is a second order differential equation, there are two constants of integration and the general solution is

( ) += tAx cos

where A is the maximum displacement, is a phase angle, and is the angular velocity and is given by

mk

=

A and are determined from the initial, boundary, conditions


Phase Angle


Simple Harmonic Motion The period of the motion is related to by

kmT

22 ==

Note that the period of the motion is independent of the displacement!

The velocity of the particle is given by

( ) +== tAdtdxv sin

with the maximum velocity being A


Simple Harmonic Motion We have so far that

( ) += tAx cos and ( ) +== tAdtdxv sin

The constants A and are determined from the initial conditions, that is what are x and v at a specified time or some other suitable initial conditions The maximum displacement occurs when the velocity is zero, which occurs at the extreme of motion (two locations: x = A)

The maximum velocity occurs when the displacement from the equilibrium position is zero (two values: v = )


SHM Energy Considerations We assume that the system is isolated and frictionless

With the force being given by xkF =It can be shown that there is a potential energy in the system that is 2

21 xkU =

The total energy is then given by

2221

21 xkvm

UKEETotal

+=

+=


SHM Energy Considerations If we substitute for x and v we find that

221 AkETotal =

The total energy is constant

The energy shifts back and forth between the kinetic energy and the potential energy, but with the sum of the two being constant


SHM Energy Considerations The relationship between kinetic energy, potential energy, displacement, velocity and acceleration can be seen in the following diagram


Choice of Oscillatory Function Note that we used the cosine function in our development of Simple Harmonic Motion

We could have also used the sine function for our description of SHM

The only difference is in the phase angle

( ) += tAx cos

( )'sin += tAx


Simple Harmonic Motion Simple Harmonic Motion can be used to describe motion in many situations under appropriate approximations

Any potential energy function that under appropriate circumstances can be approximated by a parabolic function will exhibit SHM


Simple Pendulum Consider a mass m suspended from a massless, unstretchable string of length L The forces acting on the mass are as shown The restoring force is the one perpendicular to the string

sinmgFrestoring =But this is a nonlinear function of However for small angles sinWe then have that

mgFrestoring =


Simple Pendulum From before we have that mgFrestoring =

We also have that

Lmgk =

this is then becomes the equation of SHM

Lx=

This then yields that xL

mgFrestoring =

If we then let

The frequency of oscillation is given by Lgf

21

=

which is independent of the mass attached to the string


Damped Motion Real life situations have dissipative forces

The dissipative force is often related to the velocity that is in the motion

The fact that there is a dissipative force, leads to motion that is damped, that is the amplitude decreases with time

bvFdiss =

The total force is then the sum of the restoring force and this dissipative force

vbxkFnet =

The minus sign indicates that this force is in the opposite direction to the velocity


Damped Motion This net force leads to a slightly more complicated second order differential equation

022

=++ kxdtdxb

dtxdm

The exact solution to this equation depends upon the damping constant

There are three possible solutions

Underdamped: kmb 2


Damped Motion - Underdamped kmb 2

The general solution for this situation is given by ( ) ( )tttmb eAeAex 22 212/ +=

This solution looks like

with mk

mb

=4

2

2


Damped Motion The most efficient damping, as far getting back to zero amplitude, is the critically damped case

Note that the overdamped case may yield some interesting behavior depending on the relative values of the parameters


Forced Oscillation It is also possible to drive a system such as an oscillator with an external force that is also time varying

The general differential equation is of the form

tFbvxkdt

xdm dcosmax22

=++

where the term on the right hand side is the driving force

This solution to this equation involves two functions, a complementary function and a particular solution


Forced Oscillation The solution is given by

( )( )

+= t

bmk

Ftx ddd

cos)(2222

max

with ( )

= 2

1

/2/2tan

d

dmk

mb

represents the phase difference between the driving force and the resulting motion

There can be a delay between the action of the driving force and the response of the system


Resonance

( ) 2222max

dd bmk

FA +

=

The term in front of the cosine function represents the amplitude

If we vary the angular velocity of the driving force, we find that the system has its maximum amplitude when

mk

d

This phenomenon of the amplitude peaking at a driving frequency that is near the natural frequency of the system is known as resonance


Resonance The strength of the amplitude depends upon the magnitude of the damping coefficient

The smaller the value of b the more pronounced the peak

The larger the damping the peak becomes broader, less sharp, and shifts to lower frequencies

If mkb 2> the peak disappears completely

Periodic MotionorOscillationsPeriodic MotionCharacteristicsSimple Harmonic MotionSimple Harmonic MotionPhase AngleSimple Harmonic MotionSimple Harmonic MotionSHM Energy ConsiderationsSHM Energy ConsiderationsSHM Energy ConsiderationsChoice of Oscillatory FunctionSimple Harmonic MotionSimple PendulumSimple PendulumDamped MotionDamped MotionDamped Motion - UnderdampedDamped Motion- Critically DampedDamped Motion - OverdampedDamped MotionForced OscillationForced OscillationResonanceResonance

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Lecture 01