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 $\theta_0\left(\theta_0 \ll 1\right)$ with the $x$-axis at $x=L$. If $y(x)$ is the height of the surface then the equation for $y(x)$ is:

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

An ideal fluid is flowing in a non-uniform cross-sectional tube $$X Y$$ (as shown in the figure) from end $$X$$ to end $$Y$$. If $$K_1$$ and $$K_2$$ are the kinetic energy per unit volume of the fluid at $$X$$ and $$Y$$ respectively, then the correct option is :

The maximum elongation of a steel wire of $$1 \mathrm{~m}$$ length if the elastic limit of steel and its Young's modulus, respectively, are $$8 \times 10^8 \mathrm{~N} \mathrm{~m}^{-2}$$ and $$2 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$$, is:

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

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

The amount of elastic potential energy per unit volume (in SI unit) of a steel wire of length $$100 \mathrm{~cm}$$ to stretch it by $$1 \mathrm{~mm}$$ is (if Young's modulus of the wire $$=2.0 \times 10^{11} \mathrm{Nm}^{-2}$$ ) :

Which of the following statement is not true?

The viscous drag acting on a metal sphere of diameter $$1 \mathrm{~mm}$$, falling through a fluid of viscosity $$0.8 \mathrm{~Pa}$$ s with a velocity of $$2 \mathrm{~m} \mathrm{~s}^{-1}$$ is equal to :

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

The venturi-meter works on :

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 :

Two copper vessels A and B have the same base area but of different shapes. A takes twice the volume of water as that B requires to fill upto a particular common height. Then the correct statement among the following is :

The terminal velocity of a copper ball of radius 5 mm falling through a tank of oil at room temperature is 10 cm s^$$-$$1. If the viscosity of oil at room temperature is 0.9 kg m^$$-$$1 s^$$-$$1, the viscous drag force is :

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

If a soap bubble expands, the pressure inside the bubble

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: