1.

A cylinder isotropic solid of coefficient of thermal expansion alpha and density rho (at STP) floats in a liquid of coefficient of volume expansion gamma and density d (at STP) as shown in the diagram.

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Solution :(A) Buoyant force = Mg = constant`=V_(sub)=(Mg)/(rho_(l)g)` . VOLUME of DISPLACE fluid = constant `therefore` density of fluid must be constant.
(B) `(V_("solid")-V_("sub"))="constant"rArr(V_("solid")-(M_("solid"))/(rho_("liquid")))="constant"Vxx3alphaxxDeltaT=(MgammaDeltaT)/(d)rArr3alpha(d)/(rho)`
(C ) `Ah_("in")d_("lquid")=A(h_("in")+h_("out"))rho_("solid")=M`(mass of soild)
`h_("out")=-(M)/(Arho_("soild")),h_("in")=(M)/(Arho_("soild"))-(M)/(Ad_("liquid"))=`constant
`(M(1+3alphaDeltaT))/(A(1+2alphaDeltaT)rho)-(M(1+gammaDeltaT))/(A(1+2alphaDeltaT)d)=(M)/(Arho)-(M)/(Ad),gamma=2alpha+alpha(d)/(rho)`
(D)`Ah_("in")d_("liquid")g=`Buoyant force = constant = Mg
`A_(0)(1+2alphaDeltaT)h_("in ")(d)/(1+gammaDeltaT)"constant",h_("in")=(M)/(A_(0)d)(1+(gamma-2alpha)DeltaT)gamma=2alpha`


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