A cylindrical rod with one end in a steam chamber and the other end in ice results in melting of 0.1 gm of ice per second. If the rod is replaced by another with half the length and double the radius of the first and if the thermal conductivity of material of second rod is $\frac{1}{4}$ that of first, the rate at which ice melts in gm/sec will be -
(1) 3.2               

(2) 1.6

(3) 0.2               

(4) 0.1

One end of a copper rod of length 1.0 m and area of cross-section ${10}^{-3} {\mathrm{m}}^2$ is immersed in boiling water and the other end in ice. If the coefficient of thermal conductivity of copper is 92 cal/m-s $°\mathrm{C}$ and the latent heat of ice is $8\times {10}^4$ cal/kg, then the amount of ice which will melt in one minute is -

(a) $9.2\times {10}^{-3}$ kg         (b) $8\times {10}^{-3}$
(c) $6.9\times {10}^{-3}$              (d) $5.4\times {10}^{-3}$

An ice box used for keeping eatable cold has a total wall area of 1 ${\mathrm{metre}}^2$ and a wall thickness of 5.0 cm. The thermal conductivity of the ice box is K = 0.01 joule/metre-s-$°\mathrm{C}$. It is filled with ice at 0$°\mathrm{C}$ along with eatables on a day when the temperature is 30°C. The latent heat of fusion of ice is $334 \times {10}^3 joules/kg$. The amount of ice melted in one day is ( 1 day = 86,400 seconds ) 
(a) 776 gms            (b) 7760 gms  
(c) 11520 gms         (d) 1552 gms

Five rods of the same dimensions are arranged as shown in the figure. They have thermal conductivities ${\mathrm{K}}_1, {\mathrm{K}}_2, {\mathrm{K}}_3, {\mathrm{K}}_4$ and ${\mathrm{K}}_5$. When points A and B are maintained at different temperatures, no heat flows through the central rod if :

(1) ${\mathrm{K}}_1={\mathrm{K}}_4$ and ${\mathrm{K}}_2={\mathrm{K}}_3$

(2) ${\mathrm{K}}_1{\mathrm{K}}_4={\mathrm{K}}_2{\mathrm{K}}_3$

(3) ${\mathrm{K}}_1{\mathrm{K}}_2={\mathrm{K}}_3{\mathrm{K}}_4$

(4) $\frac{{\mathrm{K}}_1}{{\mathrm{K}}_4}=\frac{{\mathrm{K}}_2}{{\mathrm{K}}_3}$

A hot metallic sphere of radius r radiates heat. It's rate of cooling is
(1) Independent of r             

(2) Proportional to r

(3) Proportional to ${\mathrm{r}}^2$             

(4) Proportional to 1/r

A solid copper sphere (density $\mathrm{\rho }$ and specific heat capacity c) of radius r at an initial temperature 200K is suspended inside a chamber whose walls are at almost 0K. The time required (in $\mathrm{\mu }$s) for the temperature of the sphere to drop to 100 K is 
(1) $\frac{72}{7}\frac{\mathrm{r\rho c}}{\mathrm{\sigma }}$                  

(2) $\frac{7}{72}\frac{\mathrm{r\rho c}}{\mathrm{\sigma }}$

(3) $\frac{27}{7}\frac{\mathrm{r\rho c}}{\mathrm{\sigma }}$                 

(4) $\frac{7}{27}\frac{\mathrm{r\rho c}}{\mathrm{\sigma }}$

One end of a copper rod of uniform cross-section and of length 3.1 m is kept in contact with ice, and the other end with water at \(100^{\circ}\mathrm{C}\). At what point along its length should a temperature of \(200^{\circ}\mathrm{C}\) be maintained so that in steady-state, the mass of ice melting be equal to that of the steam produced in the same interval time? (Assume that the whole system is insulated from the surroundings. Latent heat of fusion of ice and vaporisation of water are 80 cal/gm and 540 cal/gm respectively)
     

A sphere and a cube of same material and same volume are heated upto same temperature and allowed to cool in the same surroundings. The ratio of the amounts of radiations emitted will be
(1) 1 : 1                 

(2) $\frac{4\mathrm{\pi }}{3}:1$

(3) ${\left(\frac{\mathrm{\pi }}{6}\right)}^{1/3}:1$         

(4) $\frac{1}{2}{\left(\frac{4\mathrm{\pi }}{3}\right)}^{\frac{2}{3}}:1$

The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x, respectively are ${\mathrm{T}}_2$ and ${\mathrm{T}}_1$ (${\mathrm{T}}_2$ > ${\mathrm{T}}_1$). The rate of heat transfer through the slab, in a steady state is $\left(\frac{\mathrm{A}\left({\mathrm{T}}_2-{\mathrm{T}}_1\right)\mathrm{K}}{\mathrm{x}}\right)\mathrm{f}$, with f which equals to -

(a) 1
(b) $\frac{1}{2}$
(c) $\frac{2}{3}$
(d) $\frac{1}{3}$

                 
1. \(\frac{r_1r_2}{r_1-r_2}\)
2. \(r_1-r_2\)
3. \(({r_1-r_2})({r_1r_2})\)
4. \(\mathrm{ln}\frac{r_2}{r_1}\)