Simple Harmonic Motion
Version 0.5 12-5-08

A – amplitude; f – frequency; W – angular frequency; P – phase constant.
X – position; V – velocity; a – acceleration

A=slider([0,5,40])

f=slider([0,3,40])

prime(f)=0.125

p=slider([-2,2,40])

W=2*pi*f,P=pi*p

function(X,s)=A*cos([W*s+P]),function(V,s)=-(W*A*sin([W*s+P])),function(a,s)=-(W^2*A*cos([W*s+P]))

Oscillating mass:

vector(x,y)=vector(function(X,n),0)

abs(y)<0.05,0<x<function(X,n)

abs(y)<0.05,function(X,n)<x<0

Superimposed phase space trace:

vector(x,y)=vector(function(X,n),function(V,n))

vector(x,y)=vector(function(X,n*t),function(V,n*t))

Displacement of mass as a function of time:

function(X,prime(x))

vector(prime(x),prime(y))=vector(n,function(X,n))

vector(prime(x),prime(y))=vector(n*t,function(X,n*t))

Phase constant. Peak is shifted by an amount P/W (left if P is positive, right if P is negative) since Acos(Wn+P)=AcosW(n+P/W).

P

abs(prime(y)-A)<0.05,0<prime(x)<-P/W

abs(prime(y)-A)<0.05,-P/W<prime(x)<0

prime(y)=function(X,prime(x))

Plot potential energy, total energy, and turning points.

m – mass; k – spring constant; E – energy

m=1,k=m*W^2,E=1/2*k*A^2

1/2*k*x^2

vector(x,y)=vector(function(X,n),1/2*k*function(X,n)^2)

y=E

abs(x)=sqrt(2*E/k)

Velocity and acceleration:

function(V,prime(x))

vector(prime(x),prime(y))=vector(n*t,function(V,n*t))

function(a,prime(x))

vector(prime(x),prime(y))=vector(n*t,function(a,n*t))

Energies:

function(U,s)=1/2*k*s^2,function(K,s)=1/2*m*s^2

function(U,function(X,prime(x)))

vector(prime(x),prime(y))=vector(n*t,function(U,function(X,n*t)))

function(K,function(V,prime(x)))

vector(prime(x),prime(y))=vector(n*t,function(K,function(V,n*t)))

vector(prime(x),prime(y))=vector(n*t,function(U,function(X,n*t))+function(K,function(V,n*t)))



Author: David A. Craig <http://web.lemoyne.edu/~craigda/>

Graph of the formula

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