Properties of Solids and Liquids

Consider a water tank shown in the figure. It has one wall at \(x = L\) and can be taken to be very wide in the \(z\) direction. When filled with a liquid of surface tension \(S\) and density \(\rho\), the liquid surface makes angle \(\theta_0(\theta_0 \ll 1)\) with the \(x\)-axis at \(x = L\). If \(y(x)\) is the height of the surface then the equation for \(y(x)\) is :

(take \(\theta(x) = \sin\theta(x) = \tan\theta(x) = \frac{dy}{dx}\), \(g\) is the acceleration due to gravity)

Three identical heat conducting rods are connected in series as shown in the figure. The rods on the sides have thermal conductivity \(2K\) while that in the middle has thermal conductivity \(K\). The left end of the combination is maintained at temperature \(3T\) and the right end at \(T\). The rods are thermally insulated from outside. In steady state, temperature at the left junction is \(T_1\) and that at the right junction is \(T_2\). The ratio \(T_1/T_2\) is

A balloon is made of a material of surface tension \(S\) and its inflation outlet (from where gas is filled in it) has small area \(A\). It is filled with a gas of density \(\rho\) and takes a spherical shape of radius \(R\). When the gas is allowed to flow freely out of it, its radius \(r\) changes from \(R\) to 0 (zero) in time \(T\). If the speed \(v(r)\) of gas coming out of the balloon depends on \(r\) as \(r^a\) and \(T \propto S^{\alpha}A^{\beta}\rho^{\gamma}R^{\delta}\) then

A thin flat circular disc of radius 4.5 cm is placed gently over the surface of water. If surface tension of water is 0.07 Nm\(^{-1}\), then the excess force required to take it away from the surface is :

The maximum elongation of a steel wire of 1 m length if the elastic limit of steel and its Young's modulus, respectively, are \(8\times10^8\) N m\(^{-2}\) and \(2\times10^{11}\) N m\(^{-2}\), is :

A metallic bar of Young's modulus, \(0.5\times10^{11}\) N m\(^{-2}\) and coefficient of linear thermal expansion \(10^{-5}\,{}^\circ\)C\(^{-1}\), length 1 m and area of cross-section \(10^{-3}\) m\(^2\) is heated from \(0^\circ\)C to \(100^\circ\)C without expansion or bending. The compressive force developed in it is:

Let a wire be suspended from the ceiling (rigid support) and stretched by a weight \(W\) attached at its free end. The longitudinal stress at any point of cross-sectional area \(A\) of the wire is :

The amount of energy required to form a soap bubble of radius 2 cm from a soap solution is nearly : (surface tension of soap solution \(=0.03\,\mathrm{N\,m^{-1}}\))

The venturi-meter works on :

If a soap bubble expands, the pressure inside the bubble :

A spherical ball is dropped in a long column of a highly viscous liquid. The curve in the graph shown, which represents the speed of the ball (\(v\)) as a function of time (t) is :

Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : The stretching of a spring is determined by the shear modulus of the material of the spring.

Reason (R) : A coil spring of copper has more tensile strength than a steel spring of same dimensions.

In the light of the above statements, choose the most appropriate answer from the options given below :

A cup of coffee cools from 90°C to 80°C in t minutes, when the room temperature is 20°C. The time taken by a similar cup of coffee to cool from 80°C to 60°C at a room temperature same at 20°C is :

The velocity of a small ball of mass M and density d, when dropped in a container filled with glycerine becomes constant after some time. If the density of glycerine is \(\frac{d}{2}\), then the viscous force acting on the ball will be :

A wire of length L, area of cross section A is hanging from a fixed support. The length of the wire changes to L\(_1\) when mass M is suspended from its free end. The expression for Young's modulus is :

The quantities of heat required to raise the temperature of two solid copper spheres of radii \(r_1\) and \(r_2\) (\(r_1 = 1.5\,r_2\)) through 1 K are in the ratio :

A capillary tube of radius r is immersed in water and water rises in it to a height h. The mass of the water in the capillary is 5 g. Another capillary tube of radius 2r is immersed in water. The mass of water that will rise in this tube is :

When a block of mass M is suspended by a long wire of length L, the length of the wire becomes \((L+l)\). The elastic potential energy stored in the extended wire is :

A soap bubble, having radius of 1 mm, is blown from a detergent solution having a surface tension of \(2.5\times10^{-2}\) N/m. The pressure inside the bubble equals at a point \(Z_0\) below the free surface of water in a container. Taking \(g = 10\) m/s\(^2\), density of water \(= 10^3\) kg/m\(^3\), the value of \(Z_0\) is :

A copper rod of 88 cm and an aluminium rod of unknown length have their increase in length independent of increase in temperature. The length of aluminium rod is : (\(\alpha_{Cu} = 1.7\times10^{-5}\) K\(^{-1}\) and \(\alpha_{Al} = 2.2\times10^{-5}\) K\(^{-1}\))

The unit of thermal conductivity is :

A small hole of area of cross-section 2 mm\(^2\) is present near the bottom of a fully filled open tank of height 2 m. Taking \(g = 10\) m/s\(^2\), the rate of flow of water through the open hole would be nearly :

The power radiated by a black body is P and it radiates maximum energy at wavelength, \(\lambda_0\). If the temperature of the black body is now changed so that it radiates maximum energy at wavelength \(\frac34\lambda_0\), the power radiated by it becomes nP. The value of n is

Two wires are made of the same material and have the same volume. The first wire has cross-sectional area A and the second wire has cross-sectional area 3A. If the length of the first wire is increased by \(\Delta l\) on applying a force F, how much force is needed to stretch the second wire by the same amount ?

A small sphere of radius 'r' falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity, is proportional to

Two rods A and B of different materials are welded together as shown in figure. Their thermal conductivities are \(K_1\) and \(K_2\). The thermal conductivity of the composite rod will be :

A spherical black body with a radius of 12 cm radiates 450 watt power at 500 K. If the radius were halved and the temperature doubled, the power radiated in watt would be :

The bulk modulus of a spherical object is 'B'. If it is subjected to uniform pressure 'p', the fractional decrease in radius is :

A U tube with both ends open to the atmosphere, is partially filled with water. Oil, which is immiscible with water, is poured into one side until it stands at a distance of 10 mm above the water level on the other side. Meanwhile the water rises by 65 mm from its original level (see diagram). The density of the oil is :

A black body is at a temperature of 5760 K. The energy of radiation emitted by the body at wavelength 250 nm is \(U_1\), at wavelength 500 nm is \(U_2\) and that at 1000 nm is \(U_3\). Wien's constant, \(b = 2.88\times10^{6}\) nmK. Which of the following is correct ?

Coefficient of linear expansion of brass and steel rods are \(\alpha_1\) and \(\alpha_2\). Lengths of brass and steel rods are \(l_1\) and \(l_2\) respectively. If \((l_2 - l_1)\) is maintained same at all temperatures, which one of the following relations holds good ?

Two non-mixing liquids of densities \(\rho\) and \(n\rho\) (n > 1) are put in a container. The height of each liquid is h. A solid cylinder of length L and density d is put in this container. The cylinder floats with its axis vertical and length \(pL\) (p < 1) in the denser liquid. The density d is equal to :

A piece of ice falls from a height h so that it melts completely. Only one - quarter of the heat produced is absorbed by the ice and all energy of ice gets converted into heat during its fall. The value of h is : [Latent heat of ice is \(3.4\times10^{5}\) J/kg and g = 10 N/kg]

A rectangular film of liquid is extended from (4 cm \(\times\) 2 cm) to (5 cm \(\times\) 4 cm). If the work done is \(3\times10^{-4}\) J, the value of the surface tension of the liquid is

Three liquids of densities \(\rho_1,\rho_2\) and \(\rho_3\) (with \(\rho_1>\rho_2>\rho_3\)), having the same value of surface tension \(T\), rise to the same height in three identical capillaries. The angles of contact \(\theta_1,\theta_2\) and \(\theta_3\) obey

Two identical bodies are made of a material for which the heat capacity increases with temperature. One of these is at 100 °C, while the other one is at 0 °C. If the two bodies are brought into contact, then, assuming no heat loss, the final common temperature is

A body cools from a temperature \(3T\) to \(2T\) in 10 minutes. The room temperature is \(T\). Assume that Newton's law of cooling is applicable. The temperature of the body at the end of next 10 minutes will be