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A battery of emf $E_0=12$ V is connected across a 4m long uniform wire having resistance $4\Omega /m$. The cells of small emf’s ${ϵ}_1=2V$ and ${ϵ}_2=4V$ having internal resistance $2\Omega$ and $6\Omega$ respectively, are connected as shown in the figure. If galvanometer shows no deflection at the point N, the distance of point N from the point A is equal to

A sphere of radius $R$ is half submerged in liquid of density $\rho$. If the sphere is slightly pushed down and released, it oscillates, then what is the frequency of its oscillation ?

Find the terminal velocity of a free falling water drop of radius $0.04\text{mm}$:-
The coefficient of viscosity of air is $1.9\times {10}^{-5}{\text{Ns/m}}^2$ and its density is $1.2{\text{kg/m}}^3$. Density of water is $1000{\text{kg/m}}^3$. Take $g=10{\text{m/s}}^2$.

The figure shows a $U-$ tube of uniform cross-sectional area $A$ having a liquid of mass $M$ and density $\rho$. If the liquid in the column is slightly displaced, then it oscillates with a time period of :
[$g=$ acceleration due to gravity]

A body starts from rest. What is the ratio of the distance travelled by the body during the $4^{th}$ and $3^{rd}$ second ?

A ball is thrown at a triangular wedge with an angle of inclination ${60}^o$ parallel to the ground as shown in the figure. After bouncing against the wedge, the velocity of the ball is perpendicular to the ground. What is the coefficient of restitution?

A wire of initial length $L$ and radius $r$ is stretched by a length $l$. Another wire of same material but with initial length $2L$ and radius $2r$ is stretched by a length $2l$. The ratio of the stored elastic energy per unit volume in the first and second wire is

A man of mass $50\text{kg}$ is pulling a rope on a plank of mass $100\text{kg}$ kept on a smooth surface as shown, with force of $100\text{N}$. If both man and plank move together, find force of friction acting on man.
Given coefficient of friction between man and plank is $\frac{1}{6}$.

A small block of mass $m$ is placed over a long plank of mass $M$. Coefficient of friction between them is $\mu$. Ground is smooth. At $t=0$, $m$ is given a velocity $v_1$ and $M$ a velocity $v_2(>v_1)$ as shown in the figure. After this $M$ is maintained at constant acceleration $a(<\mu g)$.

The variation of velocity of block as a function of time is shown as

Using differentials, find the approximate value of the following up to 3 places of decimal.
$\sqrt{49.5}$

Under constant pressure, the temperature of 3 mol of an ideal monoatomic gas is raised by 15 K. What is the change in average kinetic energy per atom?

A diatomic gas $(\gamma =1.4)$ of mass $m$ at a pressure of $2$ atmospheres is compressed adiabatically so that its temperature rises from ${27}^{\circ }\text{C}$ to ${927}^{\circ }\text{C}.$ The pressure of the gas in final state is

A block rests on a rough plane whose inclination $\theta$ to the horizontal can be varied. Which of the following graphs indicates how the frictional force $F$ between the block and the plane varies as $\theta$ is increased?

Find the minimum value of the function $y=5x^2-2x+1$.

The temperature $T$ of one mole of an ideal gas varies with its volume $V$ as $T=-\alpha V^3+\beta V^2$, where $\alpha$ and $\beta$ are positive constants. The maximum pressure of the gas during this process is

Five gas molecules chosen at random are found to have speed of $500,600,700,800$ and $900\text{m/s}$. The approximate rms speed is

consider the situation of t eprevious problem. If the mirror reflects only 64% of the light energy falling on it, what will be the ratio of the maximum to the minimum intensity in the interference pattern observed on teh screen?

A block of mass M is pulled on a smooth horizontal table by a string making an angle $\theta$ with the horizontal as shown in figure. If the acceleration of the block is 'a', find the force applied by the string and by the table on the block.

A brass rod is to support a load of $400\text{N}$. If its elastic limit is $4.0\times {10}^8{\text{N/m}}^2$, its minimum diameter must be :

A coil of area $80$$c{m}^2$ and 50 turns is rotating with 2000 revolutions per minute about an axis perpendicular to a magnetic field of $0.05$ $Tesla$. The maximum value of the e.m.f. induced in it is

A rabbit runs across a parking lot on which a set of coordinate axes has, strangely enough, been drawn.
The coordinates (meters) of the rabbit's position as functions of time t (seconds) are given by

$x=-\frac{t^2}{2}+5t+20$
And $y=t^2-10t+30$

Find the acceleration at time t = 15 s?

What is the minimum value of $F$ needed so that block begins to move upward on frictionless incline plane as shown?

A particle is released from rest at origin. It moves under the influence of a potential field, $U=x^2-3x$. Find the Kinetic energy of the particle at $x=2$.

The velocity of water in a river is $18\text{km/hr}$ at the surface. If the river is $5\text{m}$ deep, find the shearing stress between the horizontal layers of water. The viscosity of water is ${10}^{-2}\text{Poise}$.

Two cylinders $A$ and $B$ fitted with pistons contain equal amounts of an ideal diatomic gas at $300\text{K}$. The piston of $A$ is free to move, while that of $B$ is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $30\text{K}$, then the rise in temperature of the gas in $B$ is:

Give a summary of the chapter of heat transfer?

The retardation experienced by a moving motor boat after its engine is cut off, is given by $\frac{dv}{dt}=-8v^2$. If the magnitude of velocity at cut off is $v_0=3\text{m/s}$, the magnitude of the velocity $1\text{sec}$ after the cut off is (in $\text{m/s}$)

In the arrangement shown in figure, $m_A=m_B=2\text{kg}$. String is massless and pulley is frictionless. Block $B$ is resting on a smooth horizontal surface, while friction coefficient between block $A$ and $B$ is $0.4$. The maximum horizontal force $F$ that can be applied so that block $A$ does not slip over the block $B$ ($g=10\text{m}/s^2$) is

Explain why
a) a horse cannot pull a cart and run in empty space,
b) passengers are thrown forward from their seats when a speeding bus stops suddenly,
c) it is easier to pull a lawn mower than to push it,
d) a cricketer moves his hands backwards while holding a catch.

A body of mass $5\text{kg}$ is acted upon by a variable force which varies with the displacement of the body as shown in figure. What is the speed of the body when it has covered $25\text{m}$? (Assume that the body starts from rest.)

In the depletion region of an unbiased P-N junction diode there are

Using differentials, find the approximate value of the following up to 3 places of decimal.
$(0.999{)}^{\frac{1}{10}}$

A political party has to start its procession in an area where wind is blowing at a speed of 41.4 km/h and party flags on the vehicles are fluttering along north-east direction. If the procession starts with a speed of 40 km/h towards north, find the direction of flags on the vehicles.

The x coordinate of a particle moving along x axis is given in terms of time as x = (2-t) (t-6) metre. Find the displacement of the particle (in metre) from time t = 1 to t = 4 second.

Unpolarized light of intensity $32W{m}^{-2}$ passes through three polarisers such that the transmission axis of the last polarizer is perpendicular to that of the first. The intensity of final emerging light is $3W{m}^{-2}$. The angle between the transmission axes of the first two polarizers will

A person walks up a stationary escalator in $t_1$ second. If he remains stationary on the escalator, then it can take him up in $t_2$ second. If the length of the escalator is L, then how much time would it take him to walk up the moving escalator?

A horizontal disc rotating freely about a vertical axis through its centre makes $90$ revolutions per minute. A small piece of wax of mass $m$ falls vertically on the disc and sticks to it at a distance $r$ from the axis. If the number of revolutions per minute reduces to $60$, the moment of inertia of the disc is

Critical angle is

An astronomical telescope is to be designed to have a magnifying power of $50$ in normal adjustment. If the length of the tube is $102cm$, find the powers of the objective and the eyepiece.

The gravitational field in a region is given by $\overline{E}=(4\hat{i}+\hat{j})N/kg$. Work done by this field is zero when a particle is moved along the line.

A system of particles is shown in figure. Point mass particles of same mass $m$ are present at each corner of the hexagon whose each side is equal to $a$. Find the $MOI$ about an axis passing through its $COM$ and perpendicular to its plane.

Consider the Hydrogen atom to be a proton embedded in cavity of radius (Bohr radius) whose charge is neutralised by addition of electron to the cavity in a vacuum infinitely slowly.Estimate the average total energy of an electron in its ground state as the work done in the above process.Also if the magnitude of the average Kinetic energy is half of magnitude of average potential energy find the average potential energy.

Is electric current determined by voltage or current determines voltage for ohmic conductors ?

Q. A Sport teacher was training the children for March past. On their way they come across a bridge. Then the PT teacher stopped the March past Comment. Also what resonance explain.

A particle is moving with a velocity of $v=(3+6t+9t^2)\text{cm}/s$. Find out the displacement of the particle in the interval $t=5\text{s}$ to $t=8\text{s}$.