Zdrs s cikloide

Materialna točka zdrsne z vrha gladke cikloide. Kdaj jo zapusti?

veci = {1,0};
vecj={0,1};

Cikloida

vecr = a{ϕ[t]-Sin[ϕ[t]],1-Cos[ϕ[t]]}

ParametricPlot[vecr/.a-> 1,{ϕ[t],0,2π}]

Tangentni in normalni vektor

vecv = D[vecr,t]

vect = FullSimplify[vecv/(Sqrt[vecv.vecv]),{ϕ'[t]>0,a>0,0< ϕ[t]< 2π}]

dsdϕ=Simplify[Sqrt[D[vecr,ϕ[t]].D[vecr,ϕ[t]]],a>0]

dvectds=Simplify[D[vect,ϕ[t]]/dsdϕ]

Ukrivljenost

κ=Simplify[Norm[dvectds],{a>0,0< ϕ[t]< 2π}]

vecn =Simplify[ dvectds/κ,{a>0,0< ϕ[t]< 2π}]

Simplify[vecn.vecn]

veca = s''[t] vect + κ ((s'[t])^2)vecn

vecS = S vecn

Newtonova enačba

eqN = m veca -(-m g vecj + vecS)

eqt=Simplify[eqN.vect]
eqn=Simplify[eqN.vecn]

Energijska enačba

eqE=Simplify[(1/2)m ((s'[t])^2)+ m g vecr.vecj- (m g vecr.vecj/.ϕ[t]-> π)]

Točka odlepitve

slvsp=Solve[eqE== 0,s'[t]]

eqS0=Simplify[eqn/.S-> 0]

eqϕ=Simplify[eqS0/.slvsp[[2]]]

pom1=FullSimplify[eqϕ,0<ϕ[t]<2π]

Reduce[(pom1== 0 && 0<ϕ[t]<2π && g ≠ 0&& m≠ 0),ϕ[t],Reals]

FullSimplify[%]

eqϕ/.ϕ[t]-> 3π/2