A metal piece of density `8g*m^(-3)` is suspended from a wooden hook by an weightless string. The tension in the string is `56g times g.` What will be the tension in the string, if the system is immersed in a liquid at `40^(@)C`? The surrounding temperature during the experiment is `20^(@)C`. At `20^(@)C` the specific gravity of the liquid is 1.24. The coefficients of volume expansion of the liquid and the metal are `4 times 10^(-5@)C^(-1) " and " 8 times 10^(-4@)C^(-1)` respectively.

Explanation

Volume of the metal piece at `20^(@)C`,
`V_(20)=56/8=7cm^(3)`
`therefore` Volume at `40^(@)C,`
`V_(40)=V_(20)[1+8 times 10^(-4) times 20]`
`" "=7(1+8 times 10^(-4) times 20)cm^(3)`
Volume of the displaced liquid `=V_(40)`
`therefore` Mass of displaced liquid `=V_(40) times rho_(40)`
`" "[rho_(40)=` density of the liquid at `40^(@)C]`
i.e., weight of the displaced liquid `=V_(40) times rho_(40) times g`
`therefore` upthrust
`" "=V_(40) times rho_(40) times g`
`" "=7[1+8 times 10^(-4) times 20] times 1.24/(1+20 times 4 times 10^(-5)) times 980`
`" "=7 times 1.24[1+0.016][1+8 times 10^(-4)]^(-1) times 980`
`" "=7 times 1.24 times 1.016[1-0.0008] times 980`
`" "=7 times 1.24 times 1.016 times 0.9992 times 980=8.81 times 980 dyn.`
`therefore` Tension in the string
`" " =(56-8.81) times 980=4.625 times 10^(4)dyn.`