CircularPath::usage="A point P moves over the paraboloid z = + so that its projection to the plane XY runs through the circumference + = x, z = 0, at a constant angular velocity of ω rad/sec. Parameterize the trajectory of P as a function of time. Deduce that it is a flat curve and calculate its Frenet trihedron, its curvature and its torsion in each P. Obtain the normal and tangential acceleration of P at every moment. References: www2.caminos.upm.es; Personal.";
2
x
2
y
2
x
2
y
Development:
Trajectory:
In[]:=
ζ[t_]:=Cos[ωt]+,Sin[ωt],0;
1
2
1
2
1
2
Is it a flat curve?
Let’s calculate your torsion:
In[]:=
χ=
Det[{ζ'[t],ζ''[t],ζ'''[t]}]
2
Norm[ζ'[t]ζ''[t]]
Out[]=
0
It’s a flat curve because its torsion is zero.
Curvature:
In[]:=
κ=//FullSimplify
Norm[ζ'[t]ζ''[t]]
3
Norm[ζ'[t]]
2
3/2
(+)
2
Abs[Cos[tω]]
2
Abs[Sin[tω]]
2
3/2
(+)
2
[Cos[tω]]
2
[Sin[tω]]
Normal acceleration:
In[]:=
ζ''[t]//FullSimplify
a
n
1
2
2
ω
1
2
2
ω
It can be seen that the magnitude of normal acceleration is:
a
n
2
ω
1
2
2
ω
Tangential acceleration:
a
t
α
α
a
t
Frenet trihedron:
In[]:=
ResourceFunction["TangentVector"][ζ[t],t]
-(5+3Cos[2tω]),(5+3Cos[2tω]),0;T1[t_]:=-,,0;
2
ωSin[tω]2
ω
2
2
ωCos[tω]2
ω
2
Sin[ωt]5+3Cos[2ωt]
2
2
Cos[ωt]5+3Cos[2ωt]
In[]:=
ResourceFunction["NormalVector"][ζ[t],t]//FullSimplify
-(5+3Cos[2tω]),-Sin[tω](5+3Cos[2tω]),0;N1[t_]:=-,-,0;
2Cos[tω]
2
6
ω
2
ω
2
ω
2
6
ω
2
ω
2
ω
2
2
Cos[ωt]5+3Cos[2ωt]
2
Sin[ωt]5+3Cos[2ωt]
In[]:=
ResourceFunction["BinormalVector"][ζ[t],t]//FullSimplify
0,0,;B={0,0,1};
3
ω
6
ω
Although it is not requested, we will obtain the original trajectory on the paraboloid:
Orthogonal projection of the circumference to the paraboloid:
ℒ:Cos[ωt]+,Sin[ωt],0+t{0,0,1};t==Cos[ωt]++Sin[ωt]=;γ[t_]:=Cos[ωt]+,Sin[ωt],;
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1
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1
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2
1
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2
1
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2
Cos
tω
2
1
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2
Cos
tω
2
Animation (let’s take ω = 1):
In[]:=
SwatchLegend[{Blue,Green,Orange,Gray},{"Tangent vector","Normal vector","Binormal vector","Normal acceleration"}]Manipulateζ[t_]:=Cos[t]+,Sin[t],0;γ[t_]:=Cos[t]+,Sin[t],;T1[t_]:=-,,0;N1[t_]:=-,-,0;B={0,0,1};κ=2;=;=0;χ=0;p=Grid{Point,FrenetTrihedron(TNB),NormalAccelerationAndMagnitude},ζ[t1],{T1[t1],N1[t1],{0,0,1}},-Cos[t1],-Sin[t1],0,,FrameAll,ItemSize->15,ItemStyle->12;q=Grid[{{Torsion,Curvature,TangentialAccelerationMagnitude},{0,2,0}},FrameAll,ItemSize->15,ItemStyle->12];ShowParametricPlot3D[ζ[t],{t,0,2π},PlotStyle->{Yellow,Thick}]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],Graphics3D[{Yellow,Thick,Ball[ζ[t1],0.035]}],Graphics3DYellow,Thick,Ball,0,0,0.035,ParametricPlot3D[γ[t],{t,0,2π},PlotStyle->{Black,Thick}]/.Line[pts_,rest___]:>Tube[pts,0.015,rest],Graphics3D[{Black,Thick,Ball[γ[t1],0.035]}],Graphics3D[{Green,Point[ζ[t1]],Arrowheads[.03],Arrow[Tube[{ζ[t1],ζ[t1]+N1[t1]}]]}],Graphics3D[{Orange,Point[ζ[t1]],Arrowheads[.03],Arrow[Tube[{ζ[t1],ζ[t1]+B}]]}],Graphics3D[{Blue,Point[ζ[t1]],Arrowheads[.03],Arrow[Tube[{ζ[t1],ζ[t1]+T1[t1]}]]}],Graphics3DGray,Point[ζ[t1]],Arrowheads[.03],ArrowTubeζ[t1],ζ[t1]+-Cos[t1],-Sin[t1],0,ParametricPlot3D[{uCos[v],uSin[v],},{v,0,2π},u,0,
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2
1
2
1
2
1
2
2
Cos
t
2
2
Sin[t]5+3Cos[2t]
2
2
Cos[t]5+3Cos[2t]
2
2
Cos[t]5+3Cos[2t]
2
Sin[t]5+3Cos[2t]
a
n
1
2
a
t
1
2
1
2
1
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u
1.5
,PlotStyle->{Green,Opacity[0.1]},PerformanceGoal->"Quality"],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-1.5,1.5},{-2,1.5},{-0.5,1.5}},ViewPoint{-1.3,-2.4,1.},ImageSize900,Style["Circular path of a point:",Bold,Large],{{t1,0,"Value (t1)"},0,2π,0.00001},Delimiter,{{p,2,"Measurable Parameters"}},{{q,2,"Constant charecteristics"}},Delimiter,ControlPlacementTopOut[]=
Out[]=
Note that normal acceleration (gray vector) always points towards the center of the circumference (in yellow), which is why it is also known as centripetal acceleration. The tangential velocity coincides with the tangent vector (in blue). The original trajectory of point P on the paraboloid appears in black.