When a block of iron in mercury at `0^@C,` fraction `K_1`of its volume is submerged, while at the temperature `60^@C,` a fraction `K_2` is seen to be submerged. If the coefficient of volume expansion of iron is `gamma_(Fe)` and that of mercury is `gamma_(Hg),` then the ratio `(K_1)//(K_2)` can be expressed as

Options

A. `(1+(60gamma_(Fe)))/(1+(60gamma_(hg)))`
B. `(1-(60gamma_(Fe)))/(1+(60gamma_(Hg)))`
C. `(1+(60gamma_(Fe)))/(1-(60gamma_(Hg)))`
D. `(1+(60gamma_(Hg)))/(1+(60gamma_(Fe)))`

Answer

A

Explanation

(a) For equilibrium in case `1 at 0^@C`
`Upthrust =Wt. of body`
`:. K_1Vd_@g=Vd_1g`
`rArrK_1=(d_1)/(d_2)…..(i)`


For equilibrium in case `2 at 60^@C`
When the temperature is increased the density
will decrease.
`:. d_1'=d_1(1+gamma_(Fe)xx60)`
and `d_2'=d_2(1+gamma_(Hg)xx60)`
`Again upthrust =Wt. of body`
`:. K_2V'd_2g=V'd_1'g`
`:. K_2[(d_2)/(1+gamma_(Hg)xx60)]=(d_1)/(1+gamma_(Fe)xx60)`
`:. K_2[(1+gamma_(Fe)xx60)/(1+gamma_(Hg)xx60)]=(d_1)/(d_2) rArr (K_1)/(K_2)=(1+gamma_(Fe)xx60)/(1+gamma_(Hg)xx60)