$A$ metallic rod of cross-sectional area $9.0 \, cm^2$ and length $0.54 \, m$,with the surface insulated to prevent heat loss,has one end immersed in boiling water and the other in an ice-water mixture. The heat conducted through the rod melts the ice at the rate of $1 \, g$ for every $33 \, s$. The thermal conductivity of the rod is ....... $W m^{-1} K^{-1}$.

Two slabs have thicknesses $d_{1}$ and $d_{2}$. Their thermal conductivities are $K_{1}$ and $K_{2}$ respectively. They are in series. The free ends of the combination of these two slabs are kept at temperatures $\theta_{1}$ and $\theta_{2}$. Assume $\theta_{1} > \theta_{2}$. The temperature $\theta$ of their common junction is

Two rods are connected as shown. The rods are of the same length and same cross-sectional area. In the steady state,the temperature $(\theta)$ of the interface will be........ $^oC$

An ice cube of dimensions $60\,cm \times 50\,cm \times 20\,cm$ is placed in an insulation box of wall thickness $1\,cm$. The box keeping the ice cube at $0^{\circ}C$ temperature is brought to a room of temperature $40^{\circ}C$. The rate of melting of ice is approximately. (Latent heat of fusion of ice is $3.4 \times 10^{5}\,J\,kg^{-1}$ and thermal conductivity of insulation wall is $0.05\,W\,m^{-1\circ}C^{-1}$)

Which of the following material properties is most suitable for a cooking utensil?

Two walls of thicknesses $d_1$ and $d_2$ and thermal conductivities $k_1$ and $k_2$ are in contact. In the steady state,if the temperatures at the outer surfaces are $T_1$ and $T_2$,the temperature at the common wall is