Math 5900 – Summer 2011

Lecture 1: Simple Harmonic Oscillations

Gernot LaicherUniversity of Utah - Department of Physics & Astronomy

Newton's Second Law

tamtF

mmassonexertedforcenettF

mmassofonacceleratita

timet

Acceleration

td

tvdta

)(

mmassofvelocitytv

Velocity

td

trdtv

)(

mmassofpositiontr

Hook’s Law

xkxF

lengthdunstretchesit'tocompared

stretchedisspringawhichbylengthadditionalx

constantspringk

Consider a mass-spring system on a frictionless table:

Unstretched x F = - k x

Stretched

Compressed

The system can be described with Newton’s second law as follows:

tamtF tvdt

dmtxk )(

)()( txdt

d

dt

dmtxk

)()(2

2

txdt

dmtxk

Differential equation (“Second Order”: Contains second derivative; “Linear”: The function and its derivatives appear as powers of 1)

Solution to this differential equation:

Note: Alternatively, we could also have written the general solution in a different but equivalent form:

Reinserting solution into DE:

The amplitude A is determined by the initial conditions of the system (at “t=0”) and the resonance frequency :

The phase angle is similarly determined by these initial conditions and the resonance frequency as follows:

f: frequency of oscillation T: period of oscillation A: amplitude of oscillation

Net restoring force directly proportional to displacement

DE of that same form

Simple harmonic (sinusoidal) oscillations

In our example: : fixed by k and m

A and imposed by the initial conditions

Note: Changing is equivalent to shifting the time when t=0

Spring Constant k= 0.6 N/m

Mass m= 2 kg

Amplitude A= 1.2 m

Phase 0degrees

0.547723 s

Frequency f=0.087173 Hz

Period 11.47147 s

Example:

Energy of the oscillating mass (assuming no losses due to friction)

Elastic Potential Energy:

Total Energy:

Kinetic Energy:

Mass hanging vertically in a gravitational field:

Unstretched Static Equilibrium Release Position Bottom of Motion Somewhere

Substitution:

Substitute in DE:

x=xo x=xo-A

x=xo+A Fs=-k(xo-xu)

Fg=mg

x

x=xu

Fs=-k(x-xu)

Fg=mg

DE has solutions of the same form as for the horizontal mass-spring system. Difference: Oscillation around s = 0. This is NOT the point where the spring is un-stretched as with the horizontal mass-spring system, but rather

Compare this point with the point of static equilibrium, where Fnet=0:

Conclusion: Both horizontal and vertical spring-mass systems oscillate around their respective static equilibrium points xo.

Does a swinging pendulum (mass m on a thin string of length L) perform harmonic oscillations?

L

m

Does a swinging pendulum (mass m on a thin string of length L) perform a harmonic oscillation?

Two forces acting on mass m:

Gravitational force (Fg) Tension in the string (T)

L

m

Fg=mg

T

Does a swinging pendulum (mass m on a thin string of length L) perform a harmonic oscillation?

It is convenient to express this problem in terms of r and, where r is the distance from the hinge to the mass (r=constant).

De-compose forces in radial and tangential components

L

m

Fg=mg

T

mg cos mg sin

Does a swinging pendulum (mass m on a thin string of length L) perform a harmonic oscillation?

Radial direction: r(t) = L =constant d2/dt2 r(t) = 0

L

m

Fg=mg

T

mg cos mg sin

Does a swinging pendulum (mass m on a thin string of length L) perform a harmonic oscillation?

Tangential direction: r(t) = L =constant d2/dt2 r(t) = 0

L

L

m

Fg=mg

T

mg cos mg sin

NOT strictly a harmonic oscillator!! A function f(x) can be expanded into a Taylor series around a point xo according to the following formula:

Expand the function sin() around =0:

Small angle approximation:

Newton’s law:

Solution:

Substitute into DE:

Without small angle approximation: Period cannot be represented by a closed formula. However, it can be represented by an infinite series (no derivation provided):

Example: For an amplitude of max=15degrees=0.26rad the first correction term is only

For an amplitude of max=60degrees=1.05rad the first correction term is

Many systems in nature experience restoring forces that do not strictly lead to harmonic oscillations but can be well approximated by a harmonic oscillator for small amplitudes of oscillations.