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A cylinder of mass m and radius r rolls without slipping on a concave cylinder surface of radius R. find the natural frequency of oscillations.
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Arc CP=ArcCP

R θ=r ϕ

ϕ=R θr

Translatory displacement of center of cylinder = (Rr)θ

Total rotational displacement of cylinder = ϕ=0 = Relative displacement.

KE=(KE)Translational+(KE)rotational

12 m [(Rr)θ]2+12IG(ϕθ)2

PE=mg[(Rr)(Rr) cos θ]=mg (Rr)[1cos θ]

ddt(KE+PE)=0

ddt[12m[(Rr)θ]2+12IG(ϕθ)2+mg(Rr)(1cosθ)]=0

ddt[12(Rr)2θ2+12IG[Rθrθ]2+mg(Rr)(1cosθ)]=0

ddt[12m(Rr)2θ2+12[12mr2].1r2[Rr]2θ2+mg(Rr)(1cosθ)]=0

ddt[12m(Rr)2θ2+14m(Rr)2θ2+mg(Rr)(1cosθ)]=0

ddt[34m(Rr)2θ2+mg(Rr)(1cosθ)]=0

34m(Rr)22θθ+mg(Rr)sinθ.θ=0

32m(Rr)2θ+mg(Rr)θ=0

θ+[mg(Rr)32m(Rr)2]θ=0

θ+[2g3(Rr)]θ=0

Wn=2g2(Rr) rad/sec.

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