$A$ 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 the material of the second rod is $\frac{1}{4}$ that of the first,the rate at which ice melts in $gm/sec$ will be:

Two vessels of different materials are similar in size in every respect. The same quantity of ice filled in them gets melted in $20 \text{ minutes}$ and $30 \text{ minutes}$ respectively. The ratio of their thermal conductivities will be

The temperature difference between the ends of two cylindrical rods $A$ and $B$ of the same material is $2: 3$. In steady state,the ratio of the rates of flow of heat through the rods $A$ and $B$ is $5: 9$. If the radii of the rods $A$ and $B$ are in the ratio $1: 2$,then the ratio of lengths of the rods $A$ and $B$ is

Four rods of the same material with different radii $r$ and length $l$ are used to connect two reservoirs of heat at different temperatures. Which one will conduct the most heat?

If $K_{1}$ and $K_{2}$ are the thermal conductivities,$L_{1}$ and $L_{2}$ are the lengths,and $A_{1}$ and $A_{2}$ are the cross-sectional areas of steel and copper rods respectively,such that $\frac{K_{2}}{K_{1}}=9$,$\frac{A_{1}}{A_{2}}=2$,and $\frac{L_{1}}{L_{2}}=2$. For the arrangement shown in the figure,the value of the temperature $T$ of the steel-copper junction in the steady state will be ........... $^{\circ}C$.

Rate of flow of heat through a cylindrical rod is $H_1$. The temperatures at the ends of the rod are $T_1$ and $T_2$. If all the dimensions of the rod become double and the temperature difference remains the same,the rate of flow of heat becomes $H_2$. Then: