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Question-175059




Question Number 175059 by mr W last updated on 17/Aug/22
Commented by mr W last updated on 17/Aug/22
a solid hemisphere with mass M and  radius R is connected with a thin rod   with mass m and length L at the   center.  find the period of small oscillations.
asolidhemispherewithmassMandradiusRisconnectedwithathinrodwithmassmandlengthLatthecenter.findtheperiodofsmalloscillations.
Commented by mr W last updated on 18/Aug/22
Answered by mr W last updated on 18/Aug/22
Commented by mr W last updated on 18/Aug/22
OA=(L/2)  OB=((3R)/8)  e=OC  e+(L/2)=(M/(m+M))×((L/2)+((3R)/8))  e=(M/(m+M))×((L/2)+((3R)/8))−(L/2)>0  I_P =((mL^2 )/(12))+m(R+(L/2))^2 +((83MR^2 )/(320))+M(((5R)/8))^2   I_P =m((L^2 /3)+R^2 +RL)+((13MR^2 )/(20))
OA=L2OB=3R8e=OCe+L2=Mm+M×(L2+3R8)e=Mm+M×(L2+3R8)L2>0IP=mL212+m(R+L2)2+83MR2320+M(5R8)2IP=m(L23+R2+RL)+13MR220
Commented by mr W last updated on 18/Aug/22
Commented by mr W last updated on 18/Aug/22
I_p ((dθ/dt))^2 +(m+M)g(R−e cos θ)=E=constant  2I_p ((dθ/dt))(d^2 θ/dt^2 )+(m+M)ge sin θ (dθ/dt)=0  2I_p (d^2 θ/dt^2 )+(m+M)ge sin θ=0  for small θ, sin θ≈θ  ⇒(d^2 θ/dt^2 )+(((m+M)eg)/(2I_P ))θ=0    → d.e. of s.h.m.  ω=(√(((m+M)eg)/(2I_P )))=(1/2)(√(([4(M−m)L+3MR]g)/(2m((L^2 /3)+R^2 +RL)+((13MR^2 )/(10)))))  T=((2π)/ω)=4π(√((20m((L^2 /3)+R^2 +RL)+13MR^2 )/(10[4(M−m)L+3MR]g)))
Ip(dθdt)2+(m+M)g(Recosθ)=E=constant2Ip(dθdt)d2θdt2+(m+M)gesinθdθdt=02Ipd2θdt2+(m+M)gesinθ=0forsmallθ,sinθθd2θdt2+(m+M)eg2IPθ=0d.e.ofs.h.m.ω=(m+M)eg2IP=12[4(Mm)L+3MR]g2m(L23+R2+RL)+13MR210T=2πω=4π20m(L23+R2+RL)+13MR210[4(Mm)L+3MR]g
Commented by Tawa11 last updated on 20/Aug/22
Great sir
Greatsir

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