The length of a metal bar is $20 \ cm$ and the area of cross-section is $4 \times 10^{-4} \ m^2$. If one end of the rod is kept in ice at $0^{\circ} C$ and the other end is kept in steam at $100^{\circ} C$,the mass of ice melted in one minute is $5 \ g$. The thermal conductivity of the metal in $W \ m^{-1} \ K^{-1}$ is (Latent heat of fusion $= 80 \ cal/g$):

Three rods of identical cross-section and lengths are made of three different materials of thermal conductivity $K_{1}, K_{2},$ and $K_{3}$,respectively. They are joined together at their ends to make a long rod. One end of the long rod is maintained at $100^{\circ} C$ and the other at $0^{\circ} C$. If the joints of the rod are at $70^{\circ} C$ and $20^{\circ} C$ in steady state and there is no loss of energy from the surface of the rod,the correct relationship between $K_{1}, K_{2}$ and $K_{3}$ is:

Two spheres made of different materials have radii in the ratio $2:1$ and wall thicknesses in the ratio $1:4$. They are completely filled with ice. If the time taken to melt the ice completely in the larger sphere is $25 \ min$ and in the smaller sphere is $16 \ min$,then the ratio of the thermal conductivities of the materials of the larger and smaller spheres is ..........

$A$ cylindrical metallic rod in thermal contact with two reservoirs of heat at its two ends conducts an amount of heat $Q$ in time $t$. The metallic rod is melted and the material is formed into a rod of half the radius of the original rod. What is the amount of heat conducted by the new rod,when placed in thermal contact with the two reservoirs in time $t$?

Three rods each of length $l$ and cross-sectional area $A$ are joined in series between two heat reservoirs as shown in the figure. Their thermal conductivities are $2K$,$K$,and $\frac{K}{2}$,respectively. Assuming that the conductors are insulated from the surroundings,the temperatures $T_1$ and $T_2$ of the junctions in the steady-state condition are,respectively:

$A$ rod of length $L$ with sides fully insulated is made of a material whose thermal conductivity varies with temperature $T$ as $K = \frac{\alpha}{T}$,where $\alpha$ is a constant. The ends of the rod are kept at temperatures $T_1$ and $T_2$. The temperature $T$ at a distance $x$ from the end kept at $T_1$ is: