Properties of Matter

Potential energy as a function of r is given by $U = {A \over {{r^{10}}}} - {B \over {{r^5}}}$, where r is the interatomic distance, A and B are positive constants. The equilibrium distance between the two atoms will be :

The bulk modulus of a liquid is 3 $\times$ 10¹⁰ Nm^$-$2. The pressure required to reduce the volume of liquid by 2% is :

When a ball is dropped into a lake from a height 4.9 m above the water level, it hits the water with a velocity v and then sinks to the bottom with the constant velocity v. It reaches the bottom of the lake 4.0 s after it is dropped. The approximate depth of the lake is :

In an experiment to verify Stokes law, a small spherical ball of radius r and density $\rho$ falls under gravity through a distance h in air before entering a tank of water. If the terminal velocity of the ball inside water is same as its velocity just before entering the water surface, then the value of h is proportional to : (ignore viscosity of air)

The length of metallic wire is l₁ when tension in it is T₁. It is l₂ when the tension is T₂. The original length of the wire will be :

A solid sphere, of radius R acquires a terminal velocity v₁ when falling (due to gravity) through a viscous fluid having a coefficient of viscosity . The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, v₂, when falling through the same fluid, the ratio (v₁/v₂) equals :

A cylinder of height $20$ $m$ is completely filled with water. The velocity of efflux of water (in $m{s^{ - 1}}$) through a small hole on the side wall of the cylinder near its bottom is

Spherical balls of radius $R$ are falling in a viscous fluid of viscosity $\eta$ with a velocity $v.$ The retarding viscous force acting on the spherical ball is

If the terminal speed of a sphere of gold (density $= 19.5\,\,kg/{m^3}$) is $0.2$ $m/s$ in a viscous liquid (density $= 1.5\,\,kg/{m^3}$, find the terminal speed of a sphere of silver (density $= 10.5\,\,kg/{m^3}$) of the same size in the same liquid

A spherical solid ball of volume $V$ is made of a material of density ${\rho _1}$. It is falling through a liquid of density ${\rho _2}\left( {{\rho _2} < {\rho _1}} \right)$. Assume that the liquid applies a viscous force on the ball that is proportional to the square of its speed $v,$ i.e., ${F_{viscous}} = - k{v^2}\left( {k > 0} \right).$ The terminal speed of the ball is

Water is flowing continuously from a tap having an internal diameter $8 \times {10^{ - 3}}\,\,m.$ The water velocity as it leaves the tap is $0.4\,\,m{s^{ - 1}}$ . The diameter of the water stream at a distance $2 \times {10^{ - 1}}\,\,m$ below the tap is close to :

Two tubes of radii r₁ and r₂, and lengths l₁ and l₂ , respectively, are connected in series and a liquid flows through each of them in stream line conditions. P₁ and P₂ are pressure differences across the two tubes. If P₂ is 4P₁ and l₂ is ${{{1_1}} \over 4}$, then the radius r₂ will be equal to :

The top of a water tank is open to air and its water level is mainted. It is giving out 0.74 m³ water per minute through a circular opening of 2 cm radius in its wall. The depth of the center of the opening from the level of water in the tank is close to :

Water flows into a large tank with flat bottom at the rate of 10^–4m³s^–1. Water is also leaking out of a hole ofarea 1 cm² at its bottom. If the height of the water in the tank remains steady, then this height is -

Water from a pipe is coming at a rate of 100 litres per minute. If the radius of the pipe is 5 cm, the Reynolds number for the flow is of the order of : (density of water = 1000 kg/m³, coefficient of viscosity of water = 1mPas)

Water from a tap emerges vertically downwards with an initial speed of 1.0 ms^–1 . The cross-sectional area of the tap is 10^–4 m². Assume that the pressure is constant throughout the stream of water and that the flow is streamlined. The cross-sectional area of the stream, 0.15 m below the tap would be : (Take g = 10 ms^–2)

Two identical cylindrical vessels are kept on the ground and each contain the same liquid of density d. The area of the base of both vessels is S but the height of liquid in one vessel is x₁ and in the other, x₂ . When both cylinders are connected through a pipe of negligible volume very close to the bottom, the liquid flows from one vessel to the other until it comes to equilibrium at a new height. The change in energy of the system in the process is:

An ideal fluid flows (laminar flow) through a pipe of non-uniform diameter. The maximum and minimum diameters of the pipes are 6.4 cm and 4.8 cm, respectively. The ratio of the minimum and the maximum velocities of fluid in this pipe is :

What will be the nature of flow of water from a circular tap, when its flow rate increased from 0.18 L/min to 0.48 L/min? The radius of the tap and viscosity of water are 0.5 cm and 10^$-$3 Pa s, respectively. (Density of water : 10³ kg/m³)