A sphere of mass 266.5 g and of diameter 7 cm floats on a liquid. When the liquid is heated to `35^(@)C` the sphere starts sinking in the liquid. If the density of the liquid at `0^(@)C " is " 1.527g*cm^(-3)`, find its coefficient of volume expansion. Neglect the expansion of the sphere.

Explanation

Volume of the sphere `=4/3pi(7/2)^(3)cm^(3)`
`therefore` Volume of displaced liquid at `35^(@)C=4/3pi(7/2)^(3)cm^(3)`
If the density of the liquid at `35^(@)C` is `rho_(35)`, then the mass of the displaced liquid at `35^(@)C=4/3pi(7/2)^(3) times rho_(35)g.`
From the condition of floatation,
`" " 4/3pi(7/3)^(3) times rho_(35)=266.5 " ...(1)"`
Now, `rho_(35)=rho_(0)/(1+gamma times 35)`
`" "=1.527/(1+gamma times 35)" ...(2)"`
From equations (1) and (2), we get,
`" "1.527/(1+gamma times 35)=266.5/(4/3pi(7/2)^(3))`
or, `" "1+35gamma=(4 times 22 times (7)^(3) times 1.527)/(3 times 7 times (2)^(3) times 266.5)`
or, `" " 35gamma=1.029-1`
or, `" "gamma=0.029/35=8.28 times 10^(-4@)C^(-1).`