Question Number 104281 by ajfour last updated on 20/Jul/20

Commented byajfour last updated on 20/Jul/20

How far (s=?) can the  hemisphere slide if a block of  mass m placed atop it is given a  gentle tap towards left.  Assume no friction between block  and hemisphere, but friction  coefficient 𝛍 between ground and  hemisphere (solid).  (Also assume that hemisphere  starts sliding before block breaks  contact with the hemisphere).

Answered by mr W last updated on 20/Jul/20

Commented bymr W last updated on 21/Jul/20

ω=(dθ/dt)  v=ωr  mω^2 r=mg cos θ−N−mA sin θ  ⇒N=m(g cos θ−A sin θ−ω^2 r)  R=Mg+N cos θ  N sin θ−μR=MA  N sin θ−μ(Mg+N cos θ)=MA  N(sin θ−μ cos θ)−μMg=MA  (m/M)(g cos θ−ω^2 r−A sin θ)(sin θ−μ cos θ)−μg=A  ⇒(m/M)(g cos θ−ω^2 r)(sin θ−μ cos θ)−μg=A[1+sin θ(sin θ−μ cos θ)]  mg sin θ+mA cos θ=mr(dω/dt)  (dω/dt)=ω(dω/dθ)  ⇒A=(rω(dω/dθ)−g sin θ)(1/(cos θ))  ⇒(m/M)(cos θ−(r/g)ω^2 )(sin θ−μ cos θ)−μ     =((r/g)ω(dω/dθ)−sin θ)((1+sin θ(sin θ−μ cos θ))/(cos θ))  ⇒ω=f(θ).....not really possible to solve  ......

Commented bymr W last updated on 21/Jul/20

even for μ=0 i can′t solve this d.e.