Initializations: Arrow3D

Arrow3D[{x1_,y1_,z1_},{x2_,y2_,z2_},color_]:=Module[{line,head,headsize},
headsize=0.015*(Sqrt[(((x1-x2))^2)+(((y1-y2))^2)+(((z1-z2))^2)]);
head=Sphere[{x2,y2,z2},headsize];
line=Line[{{x1,y1,z1},{x2,y2,z2}}];
Return[Graphics3D[{color,Thick,line,head}]];
]

(Subscript[a, T]) and (Subscript[a, N]) Components of Acceleration

Example 1: Circle with Increasing Speed

Curve

Consider a path in (ℝ^3) given by the curve

r[t_]={Cos[(t^2)],Sin[(t^2)],4}

curve=ParametricPlot3D[r[t],{t,0,(Sqrt[2Pi])},PlotStyle->{Blue,Thick},AxesLabel->{"x","y","z"}]

The Components of the Acceleration

aT[t_]=Simplify[((r'[t].r''[t])/(Sqrt[r'[t].r'[t]]))]
aN[t_]=Simplify[((Sqrt[Cross[r'[t],r''[t]].Cross[r'[t],r''[t]]])/(Sqrt[r'[t].r'[t]]))]

The Unit Tangent vector T and the Unit Normal vector N are given by

v[t_]=r'[t]
T[t_]=Simplify[((v[t])/(Sqrt[v[t].v[t]]))]
NormalVector[t_]=Simplify[((D[T[t],t])/(Sqrt[D[T[t],t].D[T[t],t]]))]

Let's check that we have the components computed correctly by comparing the two accelerations:

r''[t]
aT[t]*T[t]+aN[t]*NormalVector[t]

Plot

I am using my module Arrow3D to plot vectors, which we loaded in the initializations. Notice the tangential component of teh acceleration is constant  (red), and the normal component is increasing (green). The blue represents the acceleration vector.

Manipulate[
Show[
curve,
Arrow3D[r[t],r[t]+aT[t]*T[t],Red],
Arrow3D[r[t],r[t]+aN[t]*NormalVector[t],Green],
Arrow3D[r[t],r[t]+r''[t],Darker[Blue]],
PlotRange->{{-5,5},{-5,5},{3,5}}],
{t,0,(Sqrt[2Pi])}]

Example 2: Circle with Constant Speed

Curve

Consider a path in (ℝ^3) given by the curve

r[t_]={Cos[t],Sin[t],4}

curve=ParametricPlot3D[r[t],{t,0,2Pi},PlotStyle->{Blue,Thick},AxesLabel->{"x","y","z"}]

The Components of the Acceleration

aT[t_]=Simplify[((r'[t].r''[t])/(Sqrt[r'[t].r'[t]]))]
aN[t_]=Simplify[((Sqrt[Cross[r'[t],r''[t]].Cross[r'[t],r''[t]]])/(Sqrt[r'[t].r'[t]]))]

The Unit Tangent vector T and the Unit Normal vector N are given by

v[t_]=r'[t]
T[t_]=Simplify[((v[t])/(Sqrt[v[t].v[t]]))]
NormalVector[t_]=Simplify[((D[T[t],t])/(Sqrt[D[T[t],t].D[T[t],t]]))]

Let's check that we have the components computed correctly by comparing the two accelerations:

r''[t]
aT[t]*T[t]+aN[t]*NormalVector[t]

Plot

I am using my module Arrow3D to plot vectors, which we loaded in the initializations. Notice the tangential component of the acceleration is zero, and the normal component is constant (green--actually hidden by the dark blue line). The blue represents the acceleration vector.

Manipulate[
Show[
curve,
Arrow3D[r[t],r[t]+aT[t]*T[t],Red],
Arrow3D[r[t],r[t]+aN[t]*NormalVector[t],Green],
Arrow3D[r[t],r[t]+r''[t],Darker[Blue]],
PlotRange->{{-1,1},{-1,1},{3,5}}],
{t,0,2Pi}]

Example 3: Helix with Decreasing Speed

Curve

Consider a path in (ℝ^3) given by the curve

r[t_]={Cos[(t^2)],Sin[(t^2)],t}

curve=ParametricPlot3D[r[t],{t,0,2Pi},PlotStyle->{Blue,Thick},AxesLabel->{"x","y","z"}]

The Components of the Acceleration

aT[t_]=Simplify[((r'[t].r''[t])/(Sqrt[r'[t].r'[t]]))]
aN[t_]=Simplify[((Sqrt[Cross[r'[t],r''[t]].Cross[r'[t],r''[t]]])/(Sqrt[r'[t].r'[t]]))]

The Unit Tangent vector T and the Unit Normal vector N are given by

v[t_]=r'[t]
T[t_]=Simplify[((v[t])/(Sqrt[v[t].v[t]]))]
NormalVector[t_]=Simplify[((D[T[t],t])/(Sqrt[D[T[t],t].D[T[t],t]]))]

Let's check that we have the components computed correctly by comparing the two accelerations:

r''[t]
aT[t]*T[t]+aN[t]*NormalVector[t]

Plot

Manipulate[
Show[
curve,
Arrow3D[r[t],r[t]+aT[t]*T[t],Red],
Arrow3D[r[t],r[t]+aN[t]*NormalVector[t],Green],
Arrow3D[r[t],r[t]+r''[t],Darker[Blue]],
PlotRange->{{-5,5},{-5,5},{0,2Pi}}],
{t,0,2Pi}]

Example 4: Closed Space Curve

Curve

Consider a path in (ℝ^3) given by the curve

r[t_]={Cos[t],Sin[2t],Sin[t]}

curve=ParametricPlot3D[r[t],{t,0,2Pi},PlotStyle->{Blue,Thick},AxesLabel->{"x","y","z"}]

The Components of the Acceleration

aT[t_]=Simplify[((r'[t].r''[t])/(Sqrt[r'[t].r'[t]]))]
aN[t_]=Simplify[((Sqrt[Cross[r'[t],r''[t]].Cross[r'[t],r''[t]]])/(Sqrt[r'[t].r'[t]]))]

The Unit Tangent vector T and the Unit Normal vector N are given by

v[t_]=r'[t]
T[t_]=Simplify[((v[t])/(Sqrt[v[t].v[t]]))]
NormalVector[t_]=Simplify[((D[T[t],t])/(Sqrt[D[T[t],t].D[T[t],t]]))]

Let's check that we have the components computed correctly by comparing the two accelerations:

r''[t]
aT[t]*T[t]+aN[t]*NormalVector[t]

Plot

Notice the acceleration always lies in the osculating plane. Puase the animation and examine the components, to verify they are tangent and normal to the curve.

Manipulate[
Show[
curve,
Arrow3D[r[t],r[t]+aT[t]*T[t],Red],
Arrow3D[r[t],r[t]+aN[t]*NormalVector[t],Green],
Arrow3D[r[t],r[t]+r''[t],Darker[Blue]],
PlotRange->{{-3,3},{-3,3},{-3,3}}],
{t,0,2Pi}]