Two rods, one made of aluminium and the other made of steel, having initial lengths $l_1$  and $l_2$ respectively are connected together to form a single rod of length $(l_1 + l_2)$. The coefficients of linear expansion for aluminium and steel of $\alpha_1$ and $\alpha_2$ respectively. If the length of each rod increases by the same amount when  their temperature is raised by $t^oC$, then the ratio $l_1/(l_1 + l_2)$ :-

A solid cube having certain fixed melting and boiling points takes heat from some source. The variation of temperature $\theta$ of the cube with the heat supplied $Q$ is shown in the adjoining graph. The portion $BC$ of the graph represents the conversion of

$540\  g$ of ice at $0\ ^oC$ is mixed with $540\  g$ water at $80\  ^oC$. The final temperature of the mixture is

A cylindrical metal rod of length $L_0$ is shaped into a ring with a small gap as shown. On heating the system

Explain why :

$(a)$ a body with large reflectivity is a poor emitter

$(b)$ a brass tumbler feels much colder than a wooden tray on a chilly day

$(c)$ an optical pyrometer (for measuring high temperatures) calibrated for an ideal black body radiation gives too low a value for the temperature of a red hot iron piece in the open, but gives a correct value for the temperature when the same piece is in the furnace

$(d)$ the earth without its atmosphere would be inhospitably cold

$(e)$ heating systems based on circulation of steam are more efficient in warming a building than those based on circulation of hot water