1.

A cylinder of uniform mass density rho is in equilibrium under the pressure forces acting due to two ideal liquids of density, sigma_(1) and sigma_(2). Choose the correct alternative.

Answer»

This is only POSSIBLE if `sigma_(1)gt RHO gt sigma_(2)`
`(h_(2))/(h_(1))=|(sigma_(1)-rho)/(rho-sigma_(2))|`
If the cylinder is pressed slightly towards the bottom surface, it PERFORMS periodic motion
If the cylinder is pressed slightly towards the bttom surface, it reaches at the bottom surface with velocity `sqrt(|(sigma_(1)-sigma_(2))/(rho)|(gh_(1)^(2))/h)`

Solution :`rhog Ah=(sigma_(1)Ah_(1)+sigma_(2)Ah_(2))g`
`impliessigma_(1)h_(1)+sigma_(2)h_(2)=rhoh`
`sigma_(1)h_(1)+sigma_(2)h_(2)=(h_(1)+h_(2))rho( :' h_(1)+h_(2)=h)`
`implies(h_(2))/(h_(1))=(sigma_(1)-rho)/(rho-sigma_(2))`
If the cylinder is displace down by `y`
`F_("net")=rhoAhg-(sigma_(1)A(h_(1)-y)+sigma_(2)A(h_(2)+y))g`
`impliesrhoAhg-(sigma_(1)h_(1)A+sigma_(2)h_(2)A)g+(sigma_(1)-sigma_(2))Agy=(sigma_(1)-sigma_(2))Agy`
`impliesv=sqrt(|(sigma_(1)-sigma_(2))/(rho)|(gh_(1)^(2))/h)`


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