An object kept in a large room having an air temperature of \(25^\circ \text{C}\) takes \(12\) minutes to cool from \(80^\circ \text{C}\) to \(70^\circ \text{C}.\) The time taken to cool for the same object from \(70^\circ \text{C}\) to \(60^\circ \text{C}\) would be nearly:
1. \(10\) min
2. \(12\) min
3. \(20\) min
4. \(15\) min

A deep rectangular pond of surface area A, containing water (density = \(\rho,\) specific heat capacity = \(s\)), is located in a region where the outside air temperature is at a steady value of \(-26^{\circ}\mathrm{C}\)$$. The thickness of the ice layer in this pond at a certain instant is \(x\). Taking the thermal conductivity of ice as \(k\), and its specific latent heat of fusion as \(L\), the rate of increase of the thickness of the ice layer, at this instant, would be given by:
1. \(\frac{26k}{x\rho L-4s}\)
2. \(\frac{26k}{x^2\rho L}\)
3. \(\frac{26k}{x\rho L}\)
4. \(\frac{26k}{x\rho L+4s}\)

A copper rod of \(88\) cm and an aluminium rod of an unknown length have an equal increase in their lengths independent of an increase in temperature. The length of the aluminium rod is:
\(\left(\alpha_{Cu}= 1.7\times10^{-5}~\text{K}^{-1}~\text{and}~\alpha_{Al}= 2.2\times10^{-5}~\text{K}^{-1}\right)\)
1. \(68~\text{cm}\)
2. \(6.8~\text{cm}\)
3. \(113.9~\text{cm}\)
4. \(88~\text{cm}\)

A black body at 1227 °C emits radiations with maximum intensity at a wavelength of 5000 Å. If the temperature of the body is increased by 1000 °C, the maximum intensity will be observed at:

1. 4000 Å

2. 5000 Å

3. 6000 Å

4. 3000 Å

A black body is at 727 °C. It emits energy at a rate that is proportional to:

1. $(727{)}^2$

2. $(1000{)}^4$

3. $(1000{)}^2$

4. $(727{)}^4$

Assuming the sun to have a spherical outer surface of radius r, radiating like a black body at temperature t °C, the power received by a unit surface of the earth (normal to the incident rays) at a distance R from the centre of the sun is: 
(where σ is Stefan’s constant.)

1. $\frac{4{\mathrm{\pi r}}^2{\mathrm{\sigma t}}^4}{R^2}$

2. $\frac{r^2\sigma (t+273{)}^4}{4\mathrm{\pi }{R}^2}$

3. $\frac{16{\pi }^2{r}^2{\mathrm{\sigma t}}^4}{R^2}$

4. $\frac{r^2\sigma (t+273{)}^4}{R^2}$

On a new scale of temperature, which is linear and called the \(\mathrm{W}\) scale, the freezing and boiling points of water are \(39^\circ ~\mathrm{W}\)$$and \(239^\circ ~\mathrm{W}\) respectively. What will be the temperature on the new scale corresponding to a temperature of \(39^\circ ~\mathrm{C}\) on the Celsius scale?
1. \(78^\circ ~\mathrm{C}\)
2. \(117^\circ ~\mathrm{W}\)
3. \(200^\circ ~\mathrm{W}\)
4. \(139^\circ ~\mathrm{W}\)

The two ends of a rod of length L and a uniform cross-sectional area A are kept at two temperatures T₁ and T₂ (T₁> T₂). The rate of heat transfer $\frac{\mathrm{dQ}}{\mathrm{dt}}$ through the rod in a steady state is given by:

1. $\frac{\mathrm{dQ}}{\mathrm{dt}}=\frac{\mathrm{KL}({\mathrm{T}}_1-{\mathrm{T}}_2)}{\mathrm{A}}$

2. $\frac{\mathrm{dQ}}{\mathrm{dt}}=\frac{\mathrm{K}({\mathrm{T}}_1-{\mathrm{T}}_2)}{\mathrm{LA}}$

3. $\frac{\mathrm{dQ}}{\mathrm{dt}}=\mathrm{KLA}({\mathrm{T}}_1-{\mathrm{T}}_2)$

4. $\frac{\mathrm{dQ}}{\mathrm{dt}}=\frac{\mathrm{KA}({\mathrm{T}}_1-{\mathrm{T}}_2)}{\mathrm{L}}$

A black body at \(227^{\circ}~\mathrm{C}\) radiates heat at the rate of \(7~ \mathrm{cal-cm^{-2}s^{-1}}\).  At a temperature of \(727^{\circ}~\mathrm{C}\), the rate of heat radiated in the same units will be:
1. \(60\)
2. \(50\)
3. \(112\)
4. \(80\)

The total radiant energy per unit area, normal to the direction of incidence, received at a distance \(R\) from the centre of a star of radius \(r,\) whose outer surface radiates as a black body at a temperature \(T\) K is given by: (Where \(\sigma\) is Stefan’s constant):
1. \(\frac{\sigma r^{2}T^{4}}{R^{2}}\)

2. \(\frac{\sigma r^{2}T^{4}}{4 \pi R^{2}}\)

3. \(\frac{\sigma r^{2}T^{4}}{R^{4}}\)

4. \(\frac{4\pi\sigma r^{2}T^{4}}{R^{2}}\)