Question

A hemispherical bowl of radius `R=0.1m` is rotating about its own axis (which is verticle) with an angular velocity `omega`. A particle of mass `10^(-2)kg` on the smooth inner surface of the bowl is also rotating with the same `omega`. The particle is at a height `h` from the bottom of the bowl (a) obtain the relation betweemn `h` and `omega`. what is the minimum value of `omega` needed, in order to have a non-zero value of `h`? (b) it is desired to measure `g` using this set up, by measuring `h` accurately. assuming that `R` and `Omega` are known precisely and least count in the measurement of `h` is `10^(-4)m`, what is the minimum possible error `Deltag` in the measured value of `g`? `(g=10m//s^(2))`

Answer 1

`N cos theta=mgimpliesN=(mg)/(cos theta)`
`N sin theta=m omega^2)r=momega^(2)r=momega^(2)R sin theta` `(since sin theta=r//R)`
`N=momega^(2)R`
`(mg)/(cos theta)=momega^(2)R(cos theta=(R-h)/(R ))`
`g=omega^(2)R cos theta=omega^(2)(R-h)impliesh=R-(g)/(omega^(2))`
`omega=sqrt((g)/((R-h)))`
For the non-zero value of `h,R-(g)/(omega^(2))gt0`
`Rgt(g)/(omega^(2))impliesomegagtsqrt((g)/(R ))`
`omega_(min)=sqrt((g)/(R ))=sqrt((10)/(0.1))=10 rad//s`
(b) `g=omega^(2)(R-h)`, for the given `omega` and `R`
`dg=omega^(2)(0-dh)`
`Deltag=-omega^(2)Deltah`
`(Deltag)_(min)=-Omega^(2)_(min)Deltah=-100xx10^(-4)=-10^(-2)m//s^(2)`