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A hemisphere of radius R and mass 4m is free to slide with it`s base on a smooth horizontal table. A particle of mass m is placed on the top of the hemisphere. The angular velocity of the particle relative to hemisphere at an angular displacement ¢ when velocity of hemisphere is v is A) 5v/Rcos¢ B) 2v/Rcos¢ C) 3v/Rcos¢ D) 6v/Rcos¢

A hemisphere of radius R and mass 4m is free to slide with it`s base on a smooth horizontal table. A particle of mass m is placed on the top of the hemisphere. The angular velocity of the particle relative to hemisphere at an angular displacement ¢ when velocity of hemisphere is v is A) 5v/Rcos¢ B) 2v/Rcos¢ C) 3v/Rcos¢ D) 6v/Rcos¢

Grade:12

3 Answers

lj
115 Points
6 years ago
When the hemisphere has moved right by distance xx the particle has slid down through angle θθ. There are no external forces acting horizontally so the CM remains in the same position horizontally :m(Rsinθ−x)=Mxm(Rsin⁡θ−x)=Mx mRsinθ=(M+m)xmRsin⁡θ=(M+m)xDifferentiate wrt time :mRωcosθ=(M+m)vmRωcos⁡θ=(M+m)vwhere v=x˙v=x˙ and ω=θ˙ω=θ˙. We are given that M=4mM=4m soω=5v/Rcosa
VISHAL Yadav
15 Points
4 years ago
There is external force acting therefore CM remains in the same position horizontally. Now when hemisphere suppose moves towards right by a small distance x  with velocity v, body declines by angle © theta. Therefore, m(Rsin©-x)=Mx
mRsin© = (m+M)x |||| becomes Rsin© = 5x
Diff.both sides w.r.t time.
Therefore, equation reduces to Rwcos©=5v
d©/dt =w and dx/dt = v.
Therefore, w=5v/Rcos© .
Kushagra Madhukar
askIITians Faculty 628 Points
3 years ago
Dear student,
Please find the solution to your problem.
 
There are no external forces acting horizontally so the CM remains in the same position horizontally
when hemisphere moves towards right by a small distance x  with velocity v, body declines by an angle θ.
Therefore, m(Rsinθ – x) = Mx
mRsinθ = (m + M)x
Now, M = 4m
Therefore, Rsinθ = 5x
Diff. both sides w.r.t time.
Rcosθ.dθ/dt = 5.dx/dt
dθ/dt = w and dx/dt = v
Rcosθw = 5v
Therefore, w = 5v/Rcosθ
 
Thanks and regards,
Kushagra

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