Properties of Matter

The surface of water in a water tank of cross section area $750 \mathrm{~cm}^{2}$ on the top of a house is $h \mathrm{~m}$ above the tap level. The speed of water coming out through the tap of cross section area $500 \mathrm{~mm}^{2}$ is $30 \mathrm{~cm} / \mathrm{s}$. At that instant, $\frac{d h}{d t}$ is $x \times 10^{-3} \mathrm{~m} / \mathrm{s}$. The value of $x$ will be ____________.

A certain pressure '$\mathrm{P}$' is applied to 1 litre of water and 2 litre of a liquid separately. Water gets compressed to $0.01 \%$ whereas the liquid gets compressed to $0.03 \%$. The ratio of Bulk modulus of water to that of the liquid is $\frac{3}{x}$. The value of $x$ is ____________.

A thin rod having a length of $1 \mathrm{~m}$ and area of cross-section $3 \times 10^{-6} \mathrm{~m}^{2}$ is suspended vertically from one end. The rod is cooled from $210^{\circ} \mathrm{C}$ to $160^{\circ} \mathrm{C}$. After cooling, a mass $\mathrm{M}$ is attached at the lower end of the rod such that the length of rod again becomes $1 \mathrm{~m}$. Young's modulus and coefficient of linear expansion of the rod are $2 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$ and $2 \times 10^{-5} \mathrm{~K}^{-1}$, respectively. The value of $\mathrm{M}$ is __________ $\mathrm{kg}$.

(Take $\mathrm{g=10~m~s^{-2}}$)

A metal block of base area 0.20 m$^2$ is placed on a table, as shown in figure. A liquid film of thickness 0.25 mm is inserted between the block and the table. The block is pushed by a horizontal force of 0.1 N and moves with a constant speed. IF the viscosity of the liquid is $5.0\times10^{-3}~\mathrm{Pl}$, the speed of block is ____________ $\times10^{-3}$ m/s.

A body cools from 60$^\circ$C to 40$^\circ$C in 6 minutes. If, temperature of surroundings is 10$^\circ$C. Then, after the next 6 minutes, its temperature will be ____________$^\circ$C.

A spherical drop of liquid splits into 1000 identical spherical drops. If u$_\mathrm{i}$ is the surface energy of the original drop and u$_\mathrm{f}$ is the total surface energy of the resulting drops, the (ignoring evaporation), ${{{u_f}} \over {{u_i}}} = \left( {{{10} \over x}} \right)$. Then value of x is ____________ :

As shown in the figure, in an experiment to determine Young's modulus of a wire, the extension-load curve is plotted. The curve is a straight line passing through the origin and makes an angle of 45$^\circ$ with the load axis. The length of wire is 62.8 cm and its diameter is 4 mm. The Young's modulus is found to be $x\times10^4$ Nm$^{-2}$. The value of $x$ is ___________.

A Spherical ball of radius 1mm and density 10.5 g/cc is dropped in glycerine of coefficient of viscosity 9.8 poise and density 1.5 g/cc. Viscous force on the ball when it attains constant velocity is $3696\times10^{-x}$ N. The value of $x$ is ________.

(Given, g = 9.8 m/s$^2$ and $\pi=\frac{22}{7}$)

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

Assertion (A): Clothes containing oil or grease stains cannot be cleaned by water wash.

Reason (R): Because the angle of contact between the oil/grease and water is obtuse.

In the light of the above statements, choose the correct answer from the option given below.

If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modulus of the material of the wire will :

A pressure-pump has a horizontal tube of cross sectional area $10 \mathrm{~cm}^{2}$ for the outflow of water at a speed of $20 \mathrm{~m} / \mathrm{s}$. The force exerted on the vertical wall just in front of the tube which stops water horizontally flowing out of the tube, is :

[given: density of water $=1000 \mathrm{~kg} / \mathrm{m}^{3}$]

Consider a cylindrical tank of radius $1 \mathrm{~m}$ is filled with water. The top surface of water is at $15 \mathrm{~m}$ from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of $5 \mathrm{~m}$ from the bottom. A force of $5 \times 10^{5} \mathrm{~N}$ is applied an the top surface of water using a piston. The speed of ifflux from the hole will be : (given atmospheric pressure $\mathrm{P}_{\mathrm{A}}=1.01 \times 10^{5} \mathrm{~Pa}$, density of water $\rho_{\mathrm{W}}=1000 \mathrm{~kg} / \mathrm{m}^{3}$ and gravitational acceleration $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ )

A balloon has mass of $10 \mathrm{~g}$ in air. The air escapes from the balloon at a uniform rate with velocity $4.5 \mathrm{~cm} / \mathrm{s}$. If the balloon shrinks in $5 \mathrm{~s}$ completely. Then, the average force acting on that balloon will be (in dyne).

The force required to stretch a wire of cross-section $1 \mathrm{~cm}^{2}$ to double its length will be : (Given Yong's modulus of the wire $=2 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}$)

A steel wire of length $3.2 \mathrm{~m}\left(\mathrm{Y}_{\mathrm{s}}=2.0 \times 10^{11} \,\mathrm{Nm}^{-2}\right)$ and a copper wire of length $4.4 \mathrm{~m}\left(\mathrm{Y}_{\mathrm{c}}=1.1 \times 10^{11} \,\mathrm{Nm}^{-2}\right)$, both of radius $1.4 \mathrm{~mm}$ are connected end to end. When stretched by a load, the net elongation is found to be $1.4 \mathrm{~mm}$. The load applied, in Newton, will be: $\quad\left(\right.$ Given $\pi=\frac{22}{7}$)

Two cylindrical vessels of equal cross-sectional area $16 \mathrm{~cm}^{2}$ contain water upto heights $100 \mathrm{~cm}$ and $150 \mathrm{~cm}$ respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is [Take, density of water $=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and $\mathrm{g}=10 \mathrm{~ms}^{-2}$ ] :

The area of cross section of the rope used to lift a load by a crane is $2.5 \times 10^{-4} \mathrm{~m}^{2}$. The maximum lifting capacity of the crane is 10 metric tons. To increase the lifting capacity of the crane to 25 metric tons, the required area of cross section of the rope should be :

(take $g=10 \,m s^{-2}$ )

A water drop of radius $1 \mathrm{~cm}$ is broken into 729 equal droplets. If surface tension of water is 75 dyne/ $\mathrm{cm}$, then the gain in surface energy upto first decimal place will be :

(Given $\pi=3.14$ )

A drop of liquid of density $\rho$ is floating half immersed in a liquid of density ${\sigma}$ and surface tension $7.5 \times 10^{-4}$ Ncm^$-$1. The radius of drop in $\mathrm{cm}$ will be :

(g = 10 ms^$-$2)

An air bubble of negligible weight having radius r rises steadily through a solution of density $\sigma$ at speed v. The coefficient of viscosity of the solution is given by :

A wire of length L is hanging from a fixed support. The length changes to L₁ and L₂ when masses 1 kg and 2 kg are suspended respectively from its free end. Then the value of L is equal to :

A block of metal weighing 2 kg is resting on a frictionless plane (as shown in figure). It is struck by a jet releasing water at a rate of 1 kgs^$-$1 and at a speed of 10 ms^$-$1. Then, the initial acceleration of the block, in ms^$-$2, will be :

A water drop of radius 1 $\mu$m falls in a situation where the effect of buoyant force is negligible. Co-efficient of viscosity of air is 1.8 $\times$ 10^$-$5 Nsm^$-$2 and its density is negligible as compared to that of water 10⁶ gm^$-$3. Terminal velocity of the water drop is :

(Take acceleration due to gravity = 10 ms^$-$2)

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

Assertion A : Product of Pressure (P) and time (t) has the same dimension as that of coefficient of viscosity.

Reason R : Coefficient of viscosity = ${{Force} \over {Velocity\,gradient}}$

Choose the correct answer from the options given below :

A water drop of diameter 2 cm is broken into 64 equal droplets. The surface tension of water is 0.075 N/m. In this process the gain in surface energy will be :

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 :

The velocity of a small ball of mass 'm' and density d₁, when dropped in a container filled with glycerin, becomes constant after some time. If the density of glycerin is d₂, then the viscous force acting on the ball, will be :

If p is the density and $\eta$ is coefficient of viscosity of fluid which flows with a speed v in the pipe of diameter d, the correct formula for Reynolds number R_e is :

The terminal velocity (v_t) of the spherical rain drop depends on the radius (r) of the spherical rain drop as :

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 :

A tube of length $50 \mathrm{~cm}$ is filled completely with an incompressible liquid of mass $250 \mathrm{~g}$ and closed at both ends. The tube is then rotated in horizontal plane about one of its ends with a uniform angular velocity $x \sqrt{F} \,\mathrm{rad} \,\mathrm{s}^{-1}$. If $\mathrm{F}$ be the force exerted by the liquid at the other end then the value of $x$ will be __________.

A metal wire of length $0.5 \mathrm{~m}$ and cross-sectional area $10^{-4} \mathrm{~m}^{2}$ has breaking stress $5 \times 10^{8} \,\mathrm{Nm}^{-2}$. A block of $10 \mathrm{~kg}$ is attached at one end of the string and is rotating in a horizontal circle. The maximum linear velocity of block will be _________ $\mathrm{ms}^{-1}$.

The velocity of a small ball of mass $0.3 \mathrm{~g}$ and density $8 \mathrm{~g} / \mathrm{cc}$ when dropped in a container filled with glycerine becomes constant after some time. If the density of glycerine is $1.3 \mathrm{~g} / \mathrm{cc}$, then the value of viscous force acting on the ball will be $x \times 10^{-4} \mathrm{~N}$, The value of $x$ is _________. [use $\left.\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right]$

The speed of a transverse wave passing through a string of length $50 \mathrm{~cm}$ and mass $10 \mathrm{~g}$ is $60 \mathrm{~ms}^{-1}$. The area of cross-section of the wire is $2.0 \mathrm{~mm}^{2}$ and its Young's modulus is $1.2 \times 10^{11} \mathrm{Nm}^{-2}$. The extension of the wire over its natural length due to its tension will be $x \times 10^{-5} \mathrm{~m}$. The value of $x$ is __________.

A string of area of cross-section $4 \mathrm{~mm}^{2}$ and length $0.5 \mathrm{~m}$ is connected with a rigid body of mass $2 \mathrm{~kg}$. The body is rotated in a vertical circular path of radius $0.5 \mathrm{~m}$. The body acquires a speed of $5 \mathrm{~m} / \mathrm{s}$ at the bottom of the circular path. Strain produced in the string when the body is at the bottom of the circle is _________ $ \times 10^{-5}$.

(use Young's modulus $10^{11} \mathrm{~N} / \mathrm{m}^{2}$ and $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$)

The diameter of an air bubble which was initially $2 \mathrm{~mm}$, rises steadily through a solution of density $1750 \mathrm{~kg} \mathrm{~m}^{-3}$ at the rate of $0.35 \,\mathrm{cms}^{-1}$. The coefficient of viscosity of the solution is _________ poise (in nearest integer). (the density of air is negligible).

In an experiment to determine the Young's modulus, steel wires of five different lengths $(1,2,3,4$, and $5 \mathrm{~m})$ but of same cross section $\left(2 \mathrm{~mm}^{2}\right)$ were taken and curves between extension and load were obtained. The slope (extension/load) of the curves were plotted with the wire length and the following graph is obtained. If the Young's modulus of given steel wires is $x \times 10^{11} \,\mathrm{Nm}^{-2}$, then the value of $x$ is __________.

A spherical soap bubble of radius 3 cm is formed inside another spherical soap bubble of radius 6 cm. If the internal pressure of the smaller bubble of radius 3 cm in the above system is equal to the internal pressure of the another single soap bubble of radius r cm. The value of r is ___________.

A square aluminum (shear modulus is $25 \times 10^{9}\, \mathrm{Nm}^{-2}$) slab of side $60 \mathrm{~cm}$ and thickness $15 \mathrm{~cm}$ is subjected to a shearing force (on its narrow face) of $18.0 \times 10^{4}$ $\mathrm{N}$. The lower edge is riveted to the floor. The displacement of the upper edge is ____________ $\mu$m.

A uniform heavy rod of mass $20 \mathrm{~kg}$, cross sectional area $0.4 \mathrm{~m}^{2}$ and length $20 \mathrm{~m}$ is hanging from a fixed support. Neglecting the lateral contraction, the elongation in the rod due to its own weight is $x \times 10^{-9} \mathrm{~m}$. The value of $x$ is _______________.

(Given, young modulus Y = 2 $\times$ 10¹¹ Nm^$-$2 and g = 10 ms^$-$2)

In an experiment to determine the Young's modulus of wire of a length exactly $1 \mathrm{~m}$, the extension in the length of the wire is measured as $0.4 \mathrm{~mm}$ with an uncertainty of $\pm\, 0.02 \mathrm{~mm}$ when a load of $1 \mathrm{~kg}$ is applied. The diameter of the wire is measured as $0.4 \mathrm{~mm}$ with an uncertainty of $\pm \,0.01 \mathrm{~mm}$. The error in the measurement of Young's modulus $(\Delta \mathrm{Y})$ is found to be $x \times 10^{10}\, \mathrm{Nm}^{-2}$. The value of $x$ is _________________. $\left(\right.$take $\mathrm{g}=10 \mathrm{~ms}^{-2}$ )

A wire of length $\mathrm{L}$ and radius $\mathrm{r}$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $\mathrm{F}$, its length increases by $5 \mathrm{~cm}$. Another wire of the same material of length $4 \mathrm{L}$ and radius $4 \mathrm{r}$ is pulled by a force $4 \mathrm{F}$ under same conditions. The increase in length of this wire is __________________ $\mathrm{cm}$.

The excess pressure inside a liquid drop is 500 Nm^$-$2. If the radius of the drop is 2 mm, the surface tension of liquid is x $\times$ 10^$-$3 Nm^$-$1. The value of x is _____________.

A small spherical ball of radius 0.1 mm and density 10⁴ kg m^$-$3 falls freely under gravity through a distance h before entering a tank of water. If, after entering the water the velocity of ball does not change and it continue to fall with same constant velocity inside water, then the value of h will be ___________ m.

(Given g = 10 ms^$-$2, viscosity of water = 1.0 $\times$ 10^$-$5 N-sm^$-$2).

A liquid of density 750 kgm^$-$3 flows smoothly through a horizontal pipe that tapers in cross-sectional area from A₁ = 1.2 $\times$ 10^$-$2 m² to A₂ = ${{{A_1}} \over 2}$. The pressure difference between the wide and narrow sections of the pipe is 4500 Pa. The rate of flow of liquid is ___________ $\times$ 10^$-$3 m³s^$-$1.

The area of cross-section of a large tank is 0.5 m². It has a narrow opening near the bottom having area of cross-section 1 cm². A load of 25 kg is applied on the water at the top in the tank. Neglecting the speed of water in the tank, the velocity of the water, coming out of the opening at the time when the height of water level in the tank is 40 cm above the bottom, will be ___________ cms^$-$1. [Take g = 10 ms^$-$2]

The elastic behaviour of material for linear stress and linear strain, is shown in the figure. The energy density for a linear strain of 5 $\times$ 10^$-$4 is __________ kJ/m³. Assume that material is elastic upto the linear strain of 5 $\times$ 10^$-$4.

The elongation of a wire on the surface of the earth is 10^$-$4 m. The same wire of same dimensions is elongated by 6 $\times$ 10^$-$5 m on another planet. The acceleration due to gravity on the planet will be ____________ ms^$-$2. (Take acceleration due to gravity on the surface of earth = 10 ms^$-$2)

An ideal fluid of density 800 kgm^$-$3, flows smoothly through a bent pipe (as shown in figure) that tapers in cross-sectional area from a to ${a \over 2}$. The pressure difference between the wide and narrow sections of pipe is 4100 Pa. At wider section, the velocity of fluid is ${{\sqrt x } \over 6}$ ms^$-$1 for x = ___________. (Given g = 10 ms^$-$2)