ParticleOnACircularCone::usage=" Suppose that the position function of a particle in space is given by r[t] = {t Cos[t], t Sin[t], 3 t}. Show that the particle moves on a circular cone. Find the angle between the velocity and acceleration vectors when t = 1.5. Find the tangential and normal components of the acceleration when t = 1.5. References: Internet: Openstax.org, 3.4 Movimiento en el espacio, prob.- 199; Personal. ";​​
Angle (in grades) between the velocity and acceleration vectors when t = 1.5:
In[]:=
r[t_]:={tCos[t],tSin[t],3t};​​θ=
180
π
ArcCos
r'[1.5].r''[1.5]
Norm[r'[1.5]]Norm[r''[1.5]]

Out[]=
80.1291
Tangential component of the acceleration:
In[]:=
a
T
=
r'[1.5].r''[1.5]
Norm[r'[1.5]]
Out[]=
0.428571
Normal component of the acceleration:
In[]:=
a
N
=
2
(Norm[r''[1.5]])
-
2
a
T
Out[]=
2.46299
Verification:
In[]:=
δ=ResourceFunction["TangentVector"][r[t],t]/.t1.5;​​ξ=ResourceFunction["NormalVector"][r[t],t]/.t1.5;​​a=
a
T
δ+
a
N
ξ​​r''[1.5]
Out[]=
{-2.1011,-1.35477,0.}
Out[]=
{-2.1011,-1.35477,0}
In[]:=
Manipulater[t_]:={tCos[t],tSin[t],3t};θ[t_]:=
180
π
ArcCos
r'[t].r''[t]
Norm[r'[t]]Norm[r''[t]]
;​​
a
T
[t_]:=
r'[t].r''[t]
Norm[r'[t]]
;​​
a
N
[t_]:=
2
(Norm[r''[t]])
-
2
(
a
T
[t])
;​​​​p=Grid[{{Position},{r[t1-0.01]}},FrameAll,FrameStylePurple,ItemStyle->12];​​p1=Grid[{{Velocity},{r'[t1-0.01]}},FrameAll,FrameStyleOrange,ItemStyle->12];​​p2=Grid[{{Acceleration},{r''[t1-0.01]}},FrameAll,FrameStyleGreen,ItemStyle->12];​​p3=Grid[{{Angle},{θ[t1-0.01]}},FrameAll,ItemStyle->12];​​p4=Grid[{{TangentialAcceleration,NormalAcceleration},{
a
T
[t1-0.01],
a
N
[t1-0.01]}},FrameAll,ItemStyle->12];​​​​Show[{ParametricPlot3D[r[t],{t,0,t1},PlotStyleDirective[Dashed,Thick,Black]],​​ParametricPlot3D[{vCos[u],vSin[u],3v},{u,0,t1},{v,0,t1},Mesh->None,PlotStyleDirective[Yellow,Opacity[0.3],Specularity[White,30]],PerformanceGoal"Quality"],​​Graphics3D[{Purple,Ball[r[t1-0.01],0.08]}],Graphics3D[{Orange,Point[r[t1-0.01]],Arrowheads[.05],Arrow[Tube[{r[t1-0.01],r[t1-0.01]+r'[t1-0.01]}]]}],​​Graphics3D[{Green,Point[r[t1-0.01]],Arrowheads[.05],Arrow[Tube[{r[t1-0.01],r[t1-0.01]+r''[t1-0.01]}]]}]​​​​},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,PlotRange{{-8,8},{-8,8},{0,25}}],Style["Trajectory of the particle on circular cone:",Bold,Large],{{t1,1,"Value (​
t
1
​)"},0.01,2π,0.1},Delimiter,p,3,"

r
​[t]:",Delimiter,p1,5,"

v
​[t]:",Delimiter,p2,7,"

a
​[t]:",Delimiter,p3,9,"θ[t] between

v
​[t] and

a
​[t]:",Delimiter,{{p4,11,"Components
a
T
​[t] and
a
N
​[t] :"}},ControlPlacementLeft
Out[]=
$Aborted