Question

Two rods of the same length, have radii in the ratio 3:4. Their densities are respectively 8000 and $9000kg/m^3$. Their specific heats are in the ratio of 2:3.When the same amount of heat is supplied to both,the changes in their lengths are in the ratio. (If their linear coefficients are in the ratio 5:6)

• A. 1:1
• B. 5:2
• C. 5:12
• D. 12:5

Answer 1

The correct option is B 5:2

Heat supplied: $Q=ms\Delta \theta$ where m is the mass of the rod, , s is the specific heat of the rod and $\Delta \theta$ is the temp difference.

$\Delta l=l\alpha \Delta \theta$ where $\alpha$ is the co-efficient of linear expansion and l is the length before being heated.

$m=\pi r^{2}l\rho$

Combining the above relations

$Q=\pi r^{2}l\rho s\Delta \theta$

$Q_1=Q_2$ given

$\therefore \pi r_1^2{l}_1{\rho }_1{s}_1\Delta {\theta }_1=\therefore \pi r_2^2{l}_2{\rho }_2{s}_2\Delta {\theta }_2$

$\therefore \pi r_1^2{l}_1{\rho }_1{s}_1\left(\frac{\Delta l_1}{l_1{\alpha }_1}\right)=\pi r_2^2{l}_2{\rho }_2{s}_2\left(\frac{\Delta l_2}{l_1{\alpha }_2}\right)$

Given $l_1=l_2$

$\therefore \frac{\Delta l_1}{\Delta l_2}=\frac{r_2^2{\rho }_2{s}_2}{r_1^2{\rho }_1{s}_1}\frac{{\alpha }_1}{{\alpha }_2}=(\frac{4}{3}{)}^2\frac{9000}{8000}\frac{3}{2}\frac{5}{6}=\frac{5}{2}$