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A uniform rod of length $10\text{m}$ is placed along $x$ axis as shown in figure. The distance of the centre of mass of the rod from the origin is

The molar heat capacity in a process of a diatomic gas, if it does a work of $\frac{Q}{4},$ when a heat of $Q$ is supplied to it is

Evaluate: $\int {\sin }^{2}xdx$

A vessel containing water is given a constant acceleration ‘a’ towards the right along a straight horizontal path. Which of the following diagrams in the figure represents the surface of the liquid?

A cannon ball is fired with a velocity 200 m/sec at an angle of ${60}^{\circ }$ with the horizontal. At the highest point of its flight it explodes into 3 equal fragments, one going vertically upwards with a velocity 100 m/sec, the second one falling vertically downwards with a velocity 100 m/sec. The third fragment will be moving with a velocity

Two conducting rings $P$ and $Q$ of radii $r$ and $2r$ rotate uniformly in opposite directions with centre of mass velocities $2v$ and $v$ respectively on a conducting surface $S$. There is a uniform magnetic field of magnitude B perpendicular to the plane of the rings. The potential difference between the highest points of the two rings is

If m, $m_n$ and $m_p$ are the masses of ${}_Z{X}^A$ nucleus, neutron and proton respectively

Concept of image charge makes it possible to find electric interaction in some cases where large conducting plates are kept infront of charge. For this we consider potential of conducting plate to zero and find another (image) charge which along with given charge can make potential of conducting plate zero. In this way we can find electrostatic force between charge and conducting plate by replacing conducting plate with image charge.



A charge q is kept infront of large conducting plate at a distance l. Work done to remove this charge slowly from l to infinity is

A vessel of volume $2{\text{m}}^3$ contains an ideal gas at pressure ${10}^5{\text{N/m}}^2$ and temperature $300\text{K}$. The gas is continously pumped out of this vessel at a constant volume rate $\frac{dV}{dt}=2\times {10}^{-3}{\text{m}}^3/\text{s}$ , keeping the temperature constant. The pressure of the gas being taken out equals the pressure inside the vessel. Find the time taken before half of the original gas is pumped out (in minutes).

A telescope of diameter 2m uses light of wavelength $5000\dot{A}$ for viewing stars. The minimum angular separation between two stars whose image is just resolved by this telescope is

A small block of mass $m$ rests on a smooth wedge of angle $\theta$. With what horizontal acceleration $a$ should the wedge be pulled $($as shown in the figure$)$ so that the small block falls freely?

The amplitude of a damped oscillation becomes half in $1\text{minute}$. If the amplitude after $4\text{minute}$ will be $\frac{1}{y}$ times of the original, find the value of $y$.

A particle which is constrained to move along the x-axis, is subjected to a force in the same direction which varies with the distance x of the particle from the origin as $F\left(x\right)=-kx+a{x}^3$. Here k and a are positive constants. For $x\ge 0$, the functional from of the potential energy $U_{\left(x\right)}$ of the particle is [IIT JEE 2002]

Derivative of $f\left(x\right)=x\sin \left(\sqrt{x}\right)$ is

A liquid soap bowl rests on a table in the dining car of SHATABDI Express. If the train is accelerating uniformly at rate of $\frac{g}{4}$ in the forward direction, the angle made by the surface of the soap contained in the bowl with the horizontal is $\theta$. Then,

A smooth sphere of radius R is made to translate in a straight line with a constant acceleration a. A particle kept on the top of the sphere is released from there at zero velocity with respect to the sphere. Find the speed of the particle with respect to the sphere as a function of the angle θ it slides

A solid sphere (radius = R) rolls without slipping in a cylindrical through (radius = 5R). Find the time period of small oscillations.

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A body of mass 1 kg tied to one end of string is revolved in a horizontal circle of radius 0.1 m with a speed of 3 revolution/sec, assuming the effect of gravity is negligible, then linear velocity, acceleration and tension in the string will be

If $x=a{t}^2+bt+c$, where $x$ is the distance travelled by the body in kilometers while $t$ is the time in seconds, then the unit of $c$ is

Two balls are thrown with the same speed but making complementary angle with the horizontal. If the angle of projection of one is ${60}^{\circ }$ and its maximum height is $h$, then the maximum height of the other will be

A loaded spring gun of mass M fires a shot of mass m with a velocity v at an angle of elevation $\theta$. The gun is initially at rest on a horizontal frictionless surface. After firing, the centre of mass of the gun-shot system

Magnitude of magnetic dipole moment of loop as shown in figure is? Each curve is of radius R.

Work required to rotate given square frame by ${180}^{\circ }$ can be given as

A lab assistant has to write the equation of a spring block system doing SHM. At t = 0 the block is in the condition as shown above.He writes the equation as $x=Asin(\omega t+\varphi )$. Find the value of $\varphi$?

If there is a lone charge in the universe, and it moves a certain distance, what is the change in its electrical potential energy?

A car starting from rest, accelerates at a constant rate $\alpha$ for some time after which it decelerates at a constant rate $\beta$ to come to rest. If the total time of travel is $t$, find the total distance travelled by it assuming it to travel in a straight line.

To explain his theory of electrons jumping orbit to orbit, Bohr used

A point source causes photoelectric effect from a small metal plate. Which of the following curves may represent the saturation photocurrent as a function of the distance between the source and the metal?

Two soap bubbles are blown. In first bubble, excess pressure is $4$ times that of second bubble. The ratio of the radii of first and second soap bubble is

Ultrasonic waves of frequency 4.5 MHz are used to detect tumor in soft tissues. The speed of sound in tissue is 1.5 km s${}^{-1}$ and that in air is 340 m s${}^{-1}$. Find the wavelength of this ultrasonic wave in tissue.

The equation of state of a gas is $(P+\frac{a{T}^2}{V})\times V_1^c=(RT+b)$ where a, b, c and R are constants. The isotherms can be represented by $P=A{V}^m-B{V}^n$ where A and B depend only on temperature. Then,

The density inside a solid sphere of radius a is given by $\rho$ =${\rho }_0$a/r where ${\rho }_0$ is the density at the surface and r denotes the distance from the centre. The gravitational field due to this sphere at a distance 2a form its centre is..

You might be familiar with the ideal gas equation from Chemistry, which relates the pressure and volume of an ideal gas with temperature as: $PV$ = $nRT$, where $nR$ is a constant. You are given a closed metal container of volume $V$, filled with a gas kept at a temperature $T$, and pressure $P$. If $\gamma$ is the coefficient of volume expansion for the gas, what is the pressure after the temperature is increased by $\Delta T$?

A cyclist riding the bicycle at a speed of $14\sqrt{3}m{s}^{-1}$ takes a turn around a circular road of radius $20\sqrt{3}$ m without skidding. Given $g=9.8m{s}^{-2}$, what is his inclination to the vertical m

A hemisphere ball of radius $R$ is set rotating about its axis of symmetry which is kept vertical. A small block kept in the bowl rotates with the bowl without slipping on its surface. If the surface of the bowl is smooth is smooth and the angle made by the radius through the block with the vertical is $\theta$, find the angular speed at which the bowl is rotating.

A block of ice at temperature $-{20}^{\circ }\text{C}$ is slowly heated and converted to steam at ${100}^{\circ }\text{C}$. Which of the following diagrams is most appropriate?

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.)

A body, constrained to move in the y-direction is subjected to a force given by $\overrightarrow{F}=(-2\hat{i}+15\hat{j}+6\hat{k})$ $\text{N}$. What is the work done by this force in moving the body a distance of $10\text{m}$ along the y-axis?

An ideal reversible heat engine converts one sixth of the heat input into work. If the temperature of the sink is reduced by ${62}^{\circ }C$, its efficiency is doubled. Find the temperature of the source and sink respectively (In $\text{Kelvin}$).

A convex lens of focal length f is placed at origin. Object is placed on negative x-axis anywhere between $-\infty$ to $+\infty$. Choose correct graph between 'u' and 'v'.

A bullet moving with a velocity of 200 cm/s penetrates a wooden block and comes to rest after traversing 4 cm inside it. What velocity is needed for travelling distance of 9 cm in same block?

Two samples A and B of the same gas have equal volumes and pressures. The gas in sample A is expanded isothermally to double its volume and the gas in B is expanded adiabatically to double its volume. If the work done by the gas is the same for the two cases, then $\gamma$ (specific heat) satisfies the equation

If $y=e^x\cot \left(x\right)$, then $\frac{dy}{dx}$ will be

In one average-life,

Two satellites $S_1$ and $S_2$ revolve round a planet in coplanar circular orbits in the same sense. Their periods of revolution are 1 hour and 8 hours respectively. The radius of the orbit of $S_1$ is ${10}^4\text{km}$.



When $S_2$ is closest to $S_1$, angular speed (in $rad/hr$) of $S_2$ as observed by an astronaut in $S_1$ is