The coefficient of area expansion $\beta$ of a rectangular sheet of a solid in terms of the coefficient of linear expansion $\alpha$ is:
1. $2\alpha$
2. $\alpha$
3. $3\alpha$
4. $\alpha^2$

A blacksmith fixes an iron ring on the rim of the wooden wheel of a horse cart. The diameter of the rim and the iron ring are $5.012$ m and $5.00$ m, respectively at $27^{\circ}\text{C}.$ To what temperature should the ring be heated so as to fit the rim of the wheel?
$\left(\text{Given:}~\alpha~\text{for iron}= 1.20 \times 10^{-5}~^\circ{\text{C}^{-1}}\right)$
1. $128^{\circ}\text{C}$
2. $118^{\circ}\text{C}$
3. $227^{\circ}\text{C}$
4. $218^{\circ}\text{C}$

A sphere of $0.047$ kg aluminium is placed for sufficient time in a vessel containing boiling water so that the sphere is at $100^{\circ}\text{C}$. It is then immediately transferred to a $0.14$ kg copper calorimeter containing $0.25$ kg water at $20^{\circ}\text{C}$. The temperature of water rises and attains a steady-state at $23^{\circ}\text{C}$. The specific heat capacity of aluminium is: 
(Given that: Specific heat capacity of copper calorimeter $= 0.386\times 10^{3}~\text{J kg}^{-1}\text{K}^{-1}$ and the specific heat capacity of water $s_w= 4.18\times 10^{3}~\text{J kg}^{-1}\text{K}^{-1})$
1. $1.811~\text{kJ kg}^{-1}\text{K}^{-1}$
2. $1.911~\text{kJ kg}^{-1}\text{K}^{-1}$
3. $0.811~\text{kJ kg}^{-1}\text{K}^{-1}$
4. $0.911~\text{kJ kg}^{-1}\text{K}^{-1}$

When $0.15$ kg of ice at $0^\circ \text{C}$ is mixed with $0.30$ kg of water at $50^\circ \text{C}$ in a container, the resulting temperature is $6.7^\circ \text{C}.$
The heat of fusion of ice is: ($S_{\text{water}}=4186$ J kg^–1 K^–1)
1. $3.43 \times 10^4$ Jkg^–1
2. $3.34 \times 10^4$ Jkg^–1
3. $3.34 \times 10^5$ Jkg^–1
4. $4.34 \times 10^5$ Jkg^–1

What is the temperature of the steel-copper junction in the steady-state of the system shown in the figure? The length of the steel rod = $15.0$ cm, length of the copper rod = $10.0$ cm, temperature of the furnace = $300^{\circ}\text{C}$, temperature of the other end = $0^{\circ}\text{C}$. The area of the cross section of the steel rod is twice that of the copper rod. (Thermal conductivity of steel = $50.2 ~\text{J s}^{-1}\text{m}^{-1}\text{K}^{-1};$ and of copper $= 385~\text{J s}^{-1}\text{m}^{-1}\text{K}^{-1})$.

         

An iron bar
$\left(L_{1}=0.1 \text{m} ,   A_{1}=0.02~\text{m}^{2} ,   K_{1}=79~\text{Wm}^{- 1}   \text{K}^{-1}\right)$ and a brass bar
$\left(L_{2}=0.1~\text{m} , A_{2}=0.02~\text{m}^{2} , K_{2}=109~\text{Wm}^{- 1}  \text{K}^{- 1}\right)$ are soldered end to end as shown in the figure. The free ends of the iron bar and brass bar are maintained at $373~ \text{K}$ and $273~ \text{K}$ respectively. The temperature of the junction of the two bars is:
             

1. $215~ \text{K}$ 
2. $315~ \text{K}$ 
3. $415~ \text{K}$ 
4. $115~ \text{K}$

An iron bar $K_1 = 79~\text{W m}^{-1}\text{K}^{-1}$ and a brass bar $K_2 = 109~\text{W m}^{-1}\text{K}^{-1}$ are soldered end to end as shown in the figure. The free ends of the iron bar and brass bar are maintained at $373$ K and $273$ K respectively. Then the equivalent thermal conductivity of the compound bar is:

An iron bar $\left(L_{1} = 0 . 1 ~ \text{m},   A_{1} =  0 . 02 ~\text{m}^{2}, K_{1} = 79~\text{W m}^{- 1} \text{K}^{- 1}\right)$ and a brass bar $\left(L_{2}= 0 . 1 ~\text{m}, A_2 = 0.02~\text{m}^2, K_2 = 109~\text{W m}^{-1}\text{K}^{-1}\right)$ are soldered end to end as shown in the figure. The free ends of the iron bar and brass bar are maintained at $373$ K and $273$ K respectively. The heat current through the compound bar is:

1. $916.1$ W
2. $826.1$ W
3. $926.1$ W
4. $726$ W