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Water falls from a height of $ 200 \, \text{m} $ into a pool. Calculate the rise in temperature of the water assuming no heat dissipation from the water in the pool. [Take: $ g = 10 \, \text{m}/\text{s}^{2} $, specific heat of water $ = 4200 \, \text{J}/(\text{kg} \, \text{K}) $]

The length of a light string is $ 1.4 \, \text{m} $ when the tension on it is $ 5 \, \text{N} $. If the tension increases to $ 7 \, \text{N} $, the length of the string is $ 1.56 \, \text{m} $. The original length of the string is __________ $ \text{m} $.

Two cylindrical rods A and B made of different materials, are joined in a straight line. The ratios of lengths, radii and thermal conductivities of these rods are: $ \frac{L_A}{L_B} = \frac{1}{2} $, $ \frac{r_A}{r_B} = 2 $ and $ \frac{K_A}{K_B} = \frac{1}{2} $. The free ends of rods A and B are maintained at $ 400 \, \text{K} $, $ 200 \, \text{K} $, respectively. The temperature of rods interface is ___ $ \text{K} $, when equilibrium is established.

A steel wire of length $ 2 \, \text{m} $ and Young’s modulus $ 2.0 \times 10^{11} \, \text{N m}^{-2} $ is stretched by a force. If Poisson ratio and transverse strain for the wire are $ 0.2 $ and $ 10^{-3} $ respectively, then the elastic potential energy density of the wire is $ \underline{\hspace{1cm}} \times 10^{5} $ (in SI units).

A cup of coffee cools from $ 90^\circ \text{C} $ to $ 80^\circ \text{C} $ in $ t $ minutes when the room temperature is $ 20^\circ \text{C} $. The time taken by the similar cup of coffee to cool from $ 80^\circ \text{C} $ to $ 60^\circ \text{C} $ at the same room temperature is:

A gun fires a lead bullet of temperature $ 300 \, \text{K} $ into a wooden block. The bullet having melting temperature of $ 600 \, \text{K} $ penetrates into the block and melts down. If the total heat required for the process is $ 625 \, \text{J} $, then the mass of the bullet is ______ grams.
(Latent heat of fusion of lead $ = 2.5 \times 10^{4} \, \text{J} \, \text{Kg}^{-1} $ and specific heat capacity of lead $ = 125 \, \text{J} \, \text{Kg}^{-1} \, \text{K}^{-1} $)

The fractional compression $ \left(\frac{\Delta V}{V}\right) $ of water at the depth of $ 2.5 \, \text{km} $ below the sea level is ________ $ \% $. Given, the Bulk modulus of water $ = 2 \times 10^{9} \, \text{N m}^{-2} $, density of water $ = 10^{3} \, \text{kg m}^{-3} $, acceleration due to gravity $ = g = 10 \, \text{m s}^{-2} $.

Consider a rectangular sheet of solid material of length $ l = 9 \, \text{cm} $ and width $ d = 4 \, \text{cm} $. The coefficient of linear expansion is $ \alpha = 3.1 \times 10^{-5} \, \text{K}^{-1} $ at room temperature and one atmospheric pressure. The mass of sheet $ m = 0.1 \, \text{kg} $ and the specific heat capacity $ C_v = 900 \, \text{J\kg}^{-1}\text{K}^{-1} $. If the amount of heat supplied to the material is $ 8.1 \times 10^2 \, \text{J} $ then change in area of the rectangular sheet is:

Match List - I with List - II.

List - I	List - II
(A) Heat capacity of body	(I) $ \text{J} \, \text{kg}^{-1} $
(B) Specific heat capacity of body	(II) $ \text{J} \, \text{K}^{-1} $
(C) Latent heat	(III) $ \text{J} \, \text{kg}^{-1} \text{K}^{-1} $
(D) Thermal conductivity	(IV) $ \text{J} \, \text{m}^{-1} \text{K}^{-1} \text{s}^{-1} $

Choose the correct answer from the options given below:

10.

An air bubble of radius $ 1.0 \, \text{mm} $ is observed at a depth of $ 20 \, \text{cm} $ below the free surface of a liquid having surface tension $ 0.095 \, \text{J/m}^{2} $ and density $ 10^{3} \, \text{kg/m}^{3} $. The difference between pressure inside the bubble and atmospheric pressure is ________ $ \text{N/m}^{2} $. (Take $ g = 10 \, \text{m/s}^{2} $)

11.

The volume contraction of a solid copper cube of edge length $ 10 \, \text{cm} $, when subjected to a hydraulic pressure of $ 7 \times 10^{6} \, \text{Pa} $, would be __________ $ \text{mm}^{3} $.
(Given bulk modulus of copper $ = 1.4 \times 10^{11} \, \text{Nm}^{-2} $)

12.

Two spherical bodies of same materials having radii $ 0.2 \, \text{m} $ and $ 0.8 \, \text{m} $ are placed in same atmosphere. The temperature of the smaller body is $ 800 \, \text{K} $ and temperature of the bigger body is $ 400 \, \text{K} $. If the energy radiated from the smaller body is $ E $, the energy radiated from the bigger body is (assume, effect of the surrounding temperature to be negligible),

13.

A $ 400 \, \mathrm{g} $ solid cube having an edge of length $ 10 \, \mathrm{cm} $ floats in water. How much volume of the cube is outside the water?
(Given: density of water $ = 1000 \, \mathrm{kg \, m^{-3}} $)

14.

A sample of a liquid is kept at $ 1 \, \text{atm} $. It is compressed to $ 5 \, \text{atm} $ which leads to change of volume of $ 0.8 \, \text{cm}^3 $. If the bulk modulus of the liquid is $ 2 \, \text{GPa} $, the initial volume of the liquid was _______ litre. [Take $ 1 \, \text{atm} = 10^{5} \, \text{Pa} $]

15.

An amount of ice of mass $ 10^{-3} \,\text{kg} $ and temperature $ -10^{\circ}\text{C} $ is transformed to vapour of temperature $ 110^{\circ}\text{C} $ by applying heat. The total amount of work required for this conversion is,
(Take, specific heat of ice $ = 2100 \,\text{Jkg}^{-1}\text{K}^{-1} $, specific heat of water $ = 4180 \,\text{Jkg}^{-1}\text{K}^{-1} $, specific heat of steam $ = 1920 \,\text{Jkg}^{-1}\text{K}^{-1} $, Latent heat of ice $ = 3.35 \times 10^{5} \,\text{Jkg}^{-1} $ and Latent heat of steam $ = 2.25 \times 10^{6} \,\text{Jkg}^{-1} $)

16.

Two slabs with square cross section of different materials $ (1, 2) $ with equal sides $ (l) $ and thickness $ d_{1} $ and $ d_{2} $ such that $ d_{2} = 2 d_{1} $ and $ l > d_{2} $. Considering lowest edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of deformation is $ \theta_{2} = 2 \theta_{1} $. If the shear moduli of material $ 1 $ is $ 4 \times 10^{9} \, \text{N} / \text{m}^{2} $, then shear moduli of material $ 2 $ is $ x \times 10^{9} \, \text{N} / \text{m}^{2} $, where value of $ x $ is _______.

17.

Two wires A and B are made of same material having ratio of lengths $ \dfrac{L_A}{L_B} = \dfrac{1}{3} $ and their diameters ratio $ \dfrac{d_A}{d_B} = 2 $. If both the wires are stretched using same force, what would be the ratio of their respective elongations?

18.

Three conductors of same length having thermal conductivity $ k_1 $, $ k_2 $ and $ k_3 $ are connected as shown in figure
.
Area of cross sections of $ 1^{\text{st}} $ and $ 2^{\text{nd}} $ conductor are same and for $ 3^{\text{rd}} $ conductor it is double of the $ 1^{\text{st}} $ conductor. The temperatures are given in the figure. In steady state condition, the value of $ \theta $ is ________ $ ^\circ \text{C} $.
(Given: $ k_1 = 60 \, \text{J s}^{-1} \text{ m}^{-1} \text{ K}^{-1} $, $ k_2 = 120 \, \text{J s}^{-1} \text{ m}^{-1} \text{ K}^{-1} $, $ k_3 = 135 \, \text{J s}^{-1} \text{ m}^{-1} \text{ K}^{-1} $ )

19.

A $ 3 \, \text{m} $ long wire of radius $ 3 \, \text{mm} $ shows an extension of $ 0.1 \, \text{mm} $ when loaded vertically by a mass of $ 50 \, \text{kg} $ in an experiment to determine Young's modulus. The value of Young's modulus of the wire as per this experiment is $ P \times 10^{11} \, \text{N m}^{-2} $, where the value of $ P $ is: [Take : $ g = 3\pi \, \text{m} \, /\text{s}^{2} $]

20.

A wire of length $ 10 \, \text{cm} $ and diameter $ 0.5 \, \text{mm} $ is used in a bulb. The temperature of the wire is $ 1727^\circ \text{C} $ and power radiated by the wire is $ 94.2 \, \text{W} $. Its emissivity is $ \frac{x}{8} $ where $ x = $ ________________.$(\text{Given } \sigma = 6.0 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}, \pi = 3.14 \text{ and assume that the emissivity of wire material}$ $\text{is same at all wavelength.})$

21.

During the melting of a slab of ice at $ 273 \, \text{K} $ at atmospheric pressure:

22.

A cylindrical rod of length $ 1 \, \text{m} $ and radius $ 4 \, \text{cm} $ is mounted vertically. It is subjected to a shear force of $ 10^5 \, \text{N} $ at the top. Considering infinitesimally small displacement in the upper edge, the angular displacement $ \theta $ of the rod axis from its original position would be:
(shear moduli, $ G = 10^{10} \, \text{N/m}^2 $ )

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