Week 2 Notebook

Circular Motion and its relation to Simple Harmonic Motion
Also Relation to complex numbers (Euler’s Formula)

A=1;ω=1;

Manipulate[{ListPlot[{{A Cos[ω t],A Sin[ω t]}},PlotRange->{{-A*1.1,A*1.1},{-A*1.1,A*1.1}},PlotStyle->Red,ImageSize->Medium,AspectRatio->1],ListPlot[{{Re[A E^(I ω t)],Im[A E^(I ω t)]}},PlotRange->{{-A*1.1,A*1.1},{-A*1.1,A*1.1}},PlotStyle->Red,ImageSize->Medium,AspectRatio->1]},{t,0,2π}]

Superposition of two un-equal frequency waves:

Manipulate[Plot[Cos[t]+Cos[a t],{t,0,64 π},PlotPoints->200],{a,.5,1.5}]

Pay special attention to what happens when a becomes close to 1 (when the two frequencies are close to the same frequency)

ω1=1;ω2=.99;

?Animate

Animate[Show[ListPlot[{{Re[E^(I ω1 t)+E^(I ω2 t)],Im[E^(I ω1 t)+E^(I ω2 t)]}},PlotRange->{{-2.1,2.1},{-2.1,2.1}},PlotStyle->Red],ListPlot[{{Re[E^(I ω1 t)],Im[E^(I ω1 t)]}},PlotRange->{{-2.1,2.1},{-2.1,2.1}},PlotStyle->Blue],ListPlot[{{Re[E^(I ω2 t)],Im[E^(I ω2 t)]}},PlotRange->{{-2.1,2.1},{-2.1,2.1}},PlotStyle->Green],AspectRatio->1],{t,0,50 π}]

.

?Show

Taylor Expansions

Series[Cos[θ],{θ,0,6}]

Series[Sin[θ],{θ,0,6}]

Series[E^(x),{x,0,6}]

?Series

Integrate[1/x,x]

Integrate[1/(1+Cos[x]),x]