Conduction Law: Differential Form

Q.

A hemisphere-shpaped ole has a diameter of 5.7 millimeters and a surface area of about 51 square millimeters. The radius of the mole doubls. Estimate the new surface area of the mole.

Q.

The temperatures of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x, respectively, are $T_2$ and $T_1$ ($T_2>T_1$). The rate of heat transfer through the slab, in a steady state is $\left(\frac{A(T_2-T_1)K}{x}\right)f$, with f equal to

1. 1

2. $\frac{1}{2}$

3. $\frac{2}{3}$

4. $\frac{1}{3}$

Q.

On a cold winter day, the atmospheric temperature is -$\theta$ (on Celsius scale) which is below 0${}^{\circ }$C. A cylindrical drum of height h made of a bad conductor is completely filled with water at 0 ${}^{\circ }$ C and is kept outside without any lid. Calculate the time taken for the whole mass of water to freeze. Thermal conductivity of ice is K, $\rho$ is the density and latent heat of fusion is L. Neglect expansion of water on freezing

1. $\frac{\rho L{h}^2}{2K\theta }$

2. $\frac{\rho L{h}^2}{K\theta }$

3. $\frac{\rho Lh}{K\theta }$

4. $\frac{\rho Lh}{2K\theta }$

Q. A rod of negligible heat capacity has length 20 cm, area of cross- section $1.0c{m}^2$ and thermal conductivity $200W/m^{\circ }C$. The temperature of one end is maintained at $0^{\circ }C$ and that of the other end is slowly and linearly varied from $0^{\circ }C$ to ${60}^{\circ }C$ in 10 minutes. Assuming no loss of heat through the sides find the total heat transmitted through the rod in these 10 minutes.

1. 1800 J
2. 2000 J
3. 2200 J
4. 2400 J

Q. A thin rod of negligible mass and cross-sectional area 4x10${}^{-6}{m}^2$, suspended vertically from one end, has a length of 0.5 m at 100${}^o$C. The rod is cooled to 0${}^o$C. Youngs modulus $=$ 10${}^{11}$Nm${}^{-2}$, coefficient of linear expansion $=$ 10${}^{-5}$K${}^{-1}$ and g $=10m{s}^{-2}$
The decrease in the length of the rod on cooling is:

1. 2x10${}^{-4}$m
2. 3x10${}^{-4}$m
3. 5x10${}^{-4}$m
4. 4x10${}^{-4}$m