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A closed of length l containing a liquid...


A closed of length `l` containing a liquid of variable density `rho(x)=rho_(0)(1+alphax)` find the net force exerted by the liquid on the axis of rotation. (take the cylinder to be massless and `A=` cross sectional area of cylinder)
(a). `rho_(0)Aomega^(2)l^(2)[(1)/(2)+(1)/(3)alphal]`
(b). `rho_(0)Aomega^(2)l^(2)[(1)/(2)+(2)/(3)alphal]`
(c). `rho_(0)Aomega^(2)l^(2)[(1)/(2)+alphal]`
(d). `rho_(0)Aomega^(2)l^(2)[(1)/(2)+(4)/(3)alphal]`

A

`rho_(0)Aomega^(2)l^(2)[(1)/(2)+(1)/(3)alphal]`

B

`rho_(0)Aomega^(2)l^(2)[(1)/(2)+(2)/(3)alphal]`

C

`rho_(0)Aomega^(2)l^(2)[(1)/(2)+alphal]`

D

`rho_(0)Aomega^(2)l^(2)[(1)/(2)+(4)/(3)alphal]`

Text Solution

Verified by Experts

`intdf=intdmomega^(2)x`
`rArrF=int_(0)^(l)rho_(0)(1+alphax)Adx.omega^(2)x=rho_(0)Aomega^(2)int_(0)^(l)(1+alphax)xdx=rho_(0)Aomega^(2)[(x^(2))/(2)+alpha(x^(3))/(3)]_(0)^(l)`
`F=rho_(0)Aomega^(2)l^(2)[(1)/(2)+alpha(l)/(3)]`
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