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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.
1 Answer
written 5.8 years ago by |
→ Arc CP=ArcCP
R θ=r ϕ
∴ ϕ=R θr
→ Translatory displacement of center of cylinder = (R–r)θ
→ Total rotational displacement of cylinder = ϕ=0 = Relative displacement.
→ KE=(KE)Translational+(KE)rotational
12 m [(R−r)θ]2+12IG(ϕ−θ)2
→ PE=mg[(R−r)–(R−r) cos θ]=mg (R−r)[1−cos θ]
ddt(KE+PE)=0
ddt[12m[(R−r)θ]2+12IG(ϕ−θ)2+mg(R−r)(1−cosθ)]=0
ddt[12(R−r)2θ2+12IG[Rθr−θ]2+mg(R−r)(1−cosθ)]=0
ddt[12m(R−r)2θ2+12[12mr2].1r2[R−r]2θ2+mg(R−r)(1−cosθ)]=0
ddt[12m(R−r)2θ2+14m(R−r)2θ2+mg(R−r)(1–cosθ)]=0
ddt[34m(R−r)2θ2+mg(R−r)(1−cosθ)]=0
34m(R−r)22θθ+mg(R−r)sinθ.θ=0
32m(R−r)2θ+mg(R−r)θ=0
θ+[mg(R−r)32m(R−r)2]θ=0
θ+[2g3(R−r)]θ=0
Wn=√2g2(R−r) rad/sec.