Three rods of identical area of cross-section and made from the same metal form the sides of an isosceles triangle ABC, which is right-angled at B. The points A and B are maintained at temperatures T and $\sqrt{2}\mathrm{T}$ respectively. In the steady state, the temperature of point C is ${\mathrm{T}}_{\mathrm{C}}$. Assuming that only heat conduction takes place, $\frac{{\mathrm{T}}_{\mathrm{C}}}{\mathrm{T}}$ is equal to:
1. $\frac{1}{\left(\sqrt{2}+1\right)}$
2. $\frac{3}{\left(\sqrt{2}+1\right)}$
3. $\frac{1}{2\left(\sqrt{2}-1\right)}$                  
4. $\frac{1}{\sqrt{3}\left(\sqrt{2}-1\right)}$

The only possibility of heat flow in a thermos flask is through its cork which is 75 ${\mathrm{cm}}^2$ in area and 5 cm thick. Its thermal conductivity is 0.0075 cal/cmsec$°\mathrm{C}$. The outside temperature is 40$°\mathrm{C}$ and latent heat of ice is 80 cal ${\mathrm{g}}^{-1}$. Time taken by 500 g of ice at 0$°\mathrm{C}$ in the flask to melt into water at 0$°\mathrm{C}$ is -

(a) 2.47 hr
(b) 4.27 hr
(c) 7.42 hr
(d) 4.72 hr

A sphere, a cube and a thin circular plate, all made of the same material and having the same mass are initially heated to a temperature of 1000°C. Which one of these will cool first ?
(1) Plate                     

(2) Sphere

(3) Cube                     

(4) None of these

Two identical conducting rods are first connected independently to two vessels, one containing water at 100$°\mathrm{C}$ and the other containing ice at 0$°\mathrm{C}$. In the second case, the rods are joined end to end and connected to the same vessels. Let ${\mathrm{q}}_1$ and ${\mathrm{q}}_2$ g / s be the rate of melting of ice in two cases respectively. The ratio of ${\mathrm{q}}_1$/${\mathrm{q}}_2$ is

(a) $\frac{1}{2}$                 (b) $\frac{2}{1}$
(c) $\frac{4}{1}$                 (d) $\frac{1}{4}$

A solid cube and a solid sphere of the same material have equal surface area. Both are at the same temperature 120$°\mathrm{C}$, then -

(1) Both the cube and the sphere cool down at the same rate

(2) The cube cools down faster than the sphere

(3) The sphere cools down faster than the cube

(4) Whichever is having more mass will cool down faster

Two bodies \(A\) and \(B\) have thermal emissivities of \(0.01\) and \(0.81\) respectively. The outer surface areas of the two bodies are the same. The two bodies emit total radiant power at the same rate. The wavelength \(\lambda_B\) corresponding to maximum spectral radiancy in the radiation from \(B\) is shifted from the wavelength corresponding to maximum spectral radiancy in the radiation from \(A\), by \(1.00~\mu\text{m}\). If the temperature of \(A\) is \(5802~\text{K}\):

A black body is at a temperature of 2880 K. The energy of radiation emitted by this object with wavelength between 499 nm and 500 nm is ${\mathrm{U}}_1$, between 999 nm and 1000nm is ${\mathrm{U}}_2$ and between 1499 nm and 1500 nm is ${\mathrm{U}}_3$. [Given : Wein's constant $\mathrm{b}=2.88\times {10}^6 \mathrm{nm} \mathrm{K}$}.  Then

(1) ${\mathrm{U}}_1=0$          

(2) ${\mathrm{U}}_3=0$

(3) ${\mathrm{U}}_1>{\mathrm{U}}_2$        

(4) ${\mathrm{U}}_2>{\mathrm{U}}_1$

A black metal foil is warmed by radiation from a small sphere at temperature T and at a distance d where surrounding temperature is $T_o$. It is found that the power received by the foil is P. If both the temperature and the distance are doubled, the power received by the foil will be - [Assume $T>>T_o$]
1. 16P                 

2.  4P

3. 2P                   

4. P

Three rods of same dimensions are arranged as shown in figure they have thermal conductivities ${\mathrm{K}}_1, {\mathrm{K}}_2$ and ${\mathrm{K}}_3$ The points P and Q are maintained at different temperatures for the heat to flow at the same rate along PRQ and PQ then which of the following option is correct ?

 
(1) ${\mathrm{K}}_3=\frac{1}{2}\left({\mathrm{K}}_1+{\mathrm{K}}_2\right)$

(2) ${\mathrm{K}}_3={\mathrm{K}}_1+{\mathrm{K}}_2$

(3) ${\mathrm{K}}_3=\frac{{\mathrm{K}}_1{\mathrm{K}}_2}{{\mathrm{K}}_1+{\mathrm{K}}_2}$

(4) ${\mathrm{K}}_3=2\left({\mathrm{K}}_1+{\mathrm{K}}_2\right)$

Two metallic spheres ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ are made of the same material and have identical surface finish. The mass of ${\mathrm{S}}_1$ is three times that of ${\mathrm{S}}_2$. Both the spheres are heated to the same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. The ratio of the initial rate of cooling of ${\mathrm{S}}_1$ to that of ${\mathrm{S}}_2$ is 
(1) $1/3$             

(2) ${\left(1/3\right)}^{1/3}$

(3) $1/\sqrt{3}$          

(4) $\sqrt{3}/1$