Question

A cubical block is floating inside a bath. The temperature of system is increased by small temperature $\Delta$T. It was found that the depth of submerged portion of curve does not change. Find the relation between coefficient of linear expansion$\left(\alpha \right)$ of the cube and volume expansion of liquid $\left(\gamma \right)$

• A. $\gamma =\alpha$
• B. $\gamma =2\alpha$
• C. $\gamma =3\alpha$
• D. $\gamma =4\alpha$

Answer 1

Answer: (B)

$\gamma =2\alpha$

$h$= height inside liquid

$a$= edge length

$f_s$= material density of cube

$f_l$= density of liquid

Force active on cube all $\to$ weight $=a^3{\rho }_{s}g$

$\to$ Buoyancy force = $a^{2}h{\rho }_{1}g$

In floatation condition, $a^3{\rho }_{s}g=a^{2}h{\rho }_{1}g$............(1)

When Temp is increased by $\Delta T$,

edge length changes to $a^{\prime }$

density of liquid decrease to ${\rho }_1$

$a^{\prime }=a(1+{\alpha }_s\Delta T)$............(2)

$P{r}^{\prime }=\frac{\rho l}{(1+rl\Delta T)}$..............(3)

Now, volume of displaced liquid = $a^{12h}$

and Buoyancy force = $a^{12}h\rho e^{\prime }g$

wt of cube = buoyancy force

$a^3{\rho }_{s}g=a^{12}hg\rho l$

$a^2(1+{\alpha }_s\Delta T{)}^{2}hg[{\rho }_1/(1+rl\Delta T)]$................(4)

divide (1) and (4) we get

$1+r_1\Delta T=(1+{\alpha }_s\Delta T{)}^2$

$1+r_1\Delta T\simeq (1+2{\alpha }_s\Delta T{)}^2$ (By Binomial Theorem)

$\Rightarrow r_1=2\alpha$

$B$