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Derive an expression for the gravitational field due to a uniform rod of length L and mass M at a point on its perpendicular bisector at a distance d from the centre.

A particular guitar wire is 30.0 cm long and vibrates at a frequency of 196 Hz when no finger is placed on it. The next higher notes on the scale are 220 Hz, 247 Hz, 262 Hz and 294 Hz. How far from the end of the string must the finger be placed to play these notes ?

The convex surface of a thin concavo-convex lens of glass of refractive index 1.5 has a radius of curvature 20 cm. The concave surface has a radius of curvature 60 cm. The convex side is silvered and placed on a horizontal surface as shown in figure.

(a) Where should a pin be placed on the axis so that its image is formed at the same place ?

(b) If the concave part is filled with water $(\mu =\frac{4}{3})$, find the distance through which the pin should be moved so that the image of the pin again coincides with the pin.

The masses of ${}^{11}\text{C}$ and ${}^{11}\text{B}$ are respectively 11.0114 u and 11.0093 u. Find the maximum energy a position can have in the ${\beta }^{+}-$ decay of ${}^{11}\text{C}$ to ${}^{11}\text{B}$

In one of the exercises to strengthen the wrist and fingers, a person squeezes and releases a soft rubber ball. Is the work done on the ball positive, negative or zero (i) during compression (ii) During expansion?

An airplane flying $490\text{m}$ above ground level at $100\text{m/s}$, releases a bomb. The horizontal distance it has to travel before it hits the ground is (Take $g=9.8{\text{m/s}}^2$)

A thick walled hollow sphere has outer radius $R$. It rolls down an inclined plane without slipping and its speed at the bottom is $v$. If the inclined plane is frictionless and the sphere slides down without rolling, its speed at the bottom is $5v/4$. What is the radius of gyration of the sphere?

A copper rod of length L and mass m is sliding down a smooth inclined plane of inclination $\theta$ with a constant
speed v. A current I is flowing in perpendicular to the plane of diagram inwards. A vertically upward magnetic
field $\overrightarrow{B}$ exist in this region.The magnitude of the required magnetic field $\overrightarrow{B}$ is

Which of the following statements is/are true

A solid cylinder of mass ($M=1\text{kg}$) and radius $R=0.5\text{m}$ is pivoted at its centre The axis of rotation of the cylinder is horizontal. Three small particles of mass ($m=0.1\text{kg}$) are mounted along its surface as shown in figure. The system is initially at rest.

The angular speed of cylinder, when it has rotated through ${90}^{\circ }$ in anticlockwise direction.
(Take $g=10{\text{m/s}}^2$)

A bullet of mass m moving at a speed v hits a ball of mass M kept at rest.A small part having mass 'm' breaks from the ball and sticks to the bullet.The remaining ball is found to move at a speed $v_1$ in the direction of the bullet.Find the velocity of the bullet after the collision.

A uniform solid cylinder of mass m can rotate freely about its axis which is kept horizontal. A particle of mass $m_0$ hangs from the end of a light string wound around the cylinder. When the system is allowed to move, the acceleration with which the particle descends is

The work done to get $n$ smaller equal size spherical drops from a bigger size spherical drop of water is proportional to:

The electric field in a region is given by $\overrightarrow{E}=\frac{3}{5}{E}_0\overrightarrow{j}$ with $E_0=2.0\times {10}^{3}N{C}^{-}1$. Find the flux of this field through a rectangular surface of area $0.2m^2$ parallel to the y-z plane.

A particle slides on. the surface of a fixed smooth spherce starting from the topmost point. Find the angle rotated, by the radius through the particle, when it leaves contact with the sphere.

A rod of mass m and length 2R is fixed along the diameter of a ring of same mass m and radius R as shown in figure. The combined body is rolling without slipping along x – axis. The angular momentum about z – axis is

The thickness of a glass plate is measured to be 2.17 mm, 2.17 mm and 2.18 mm at three different places. Find the average thickness of the plate from this data.

A uniform rope of mass $0.1\text{kg}$ and length $2.5\text{m}$ hangs from ceiling. The speed of transverse wave in the rope at upper end and at a point $0.5\text{m}$ distance from lower end will be:

$f\left(x\right)=\frac{1}{1+\frac{1}{x}};gx=\frac{1}{1+\frac{1}{f\left(x\right)}}\Rightarrow g^{\prime }\left(2\right)=$

Two cars $A$ and $B$ are $5\text{km}$ apart on a line joining South to North. Car $A$ further North is streaming west at $10\text{km/hr}$ and Car $B$ is streaming North at $10\text{km/hr}$. What is their closest distance of approach?

In an adiabatic process, $R=\frac{2}{3}{C}_v$. The pressure of the gas will be proportional to

A block of mass $200\text{gram}$ attached to a spring of stiffness $50\text{N/m}$ is lying on a frictionless floor as shown in figure. The block is moved to compress the spring by $8\text{cm}$ and released. If collision with the wall is elastic, find the time period of oscillations.

In photoelectric effect, the K.E. of electrons emitted from the metal surface depends upon

The diagram shows a small bead of mass m carrying charge q . The bead can freely move on the smooth fixed ring placed on a smooth horizontal plane. In the same plane a charge +Q has also been fixed as shown. The potential at the point P due to +Q is V. The velocity with which the bead should projected from the point P so that it can complete a circle should be greater than

A bird is flying towards north with a velocity 40 km/h and a train is moving with a velocity 40 km/h towards east. What is the velocity of the bird noted by a man in the train?

The current in a coil of self-inductance $2.0\text{H}$ is increasing according to equation $I=2\sin t^2\text{A}$. Find the amount of energy spent during the period when the current changes from $0$ to $2\text{A}$.

One mole of a monoatomic gas of molar mass $M$ undergoes a cyclic process as shown in the figure. Here, $\rho$ is the density and $P$ is the pressure of the gas. Find the efficiency of the cycle.

Mass of block A is $100\text{kg}$ and that of block B is $200\text{kg}$. As shown in figure, the block A is attached to a string tied to the wall. The coefficient of static friction between blocks A and B is $({\mu }_s{)}_A=0.2$ and the coefficient of static friction between block B and floor is $({\mu }_s{)}_B=0.3$. Then calculate the minimum value of force $F$ required to move the block B. $(g=10{\text{m/s}}^2)$

A particle of mass m rotates with a uniform angular speed $\omega$. It is viewed from a frame rotating about the Z-axis at a distance a from the particle, with a uniform angular speed ${\omega }_0$. The centrifugal force on the particle is

A spy report about a suspected car reads as follows. "The car moved 2.00 km towards east, made a perpendicular left turn, ran for 500 m, made a perpendicular right turn, ran for 4.00 km and stopped". Find the displacement of the car.

Two blocks $A$ and $B$ of mass $m$ and $2m$ are connected by a massless spring of force constant $k$. They are placed on a smooth horizontal plane. Spring is stretched by an amount $x$ and then released. The relative velocity of the blocks when the spring comes to its natural length is

A 2 kg block is placed over a 4 kg block and both are placed on a smooth horizontal surface. The coefficient of friction between the blocks is 0.20. Find the acceleration of the two blocks if a horizontal force of 12 N is applied to the upper block. Take g = 10 $m/s^2$.

Find the force of interaction between the $2kg$ and $1kg$ bodies.

How will you write the vector joining points P(1, 3,2) and Q(4, 2, 1)? Also write the unit vector for $\underset{PQ}{\to }$

In a U tube experiment, a column $AB$ of water is balanced by a column $CD$ of paraffin. The relative density of paraffin is

Whether is it possible that the brakes of car are so perfect that the car stops instantaneously. If not why?

$\underset{x\to 0}{lim}\left[\frac{lncosx}{\sqrt[4]{41+x^2}-1}\right]\text{is equal to}$

Planck’s constant (h), speed of light in vacuum (c) and Newton’s gravitational constant (G) are three fundamental constants. Which one of the following combinations of these has the dimension of length?

Differentiate the following functions with respect to x

$\frac{sin(ax+b)}{cos(cx+d)}$

Equal mass of three liquids are kept in three identical cylindrical vessels A, B and C. The densities are P_A<P_B<P_C. The force on the base will be

In the diagram particle A moves along the line y = 25$\sqrt{3}$ m with a constant velocity $\overrightarrow{v}$ of magnitude 5 m/s and parallel to the x axis. At the instant particle A passes the y axis, particle B leaves the origin with a zero initial speed and a constant acceleration $\overrightarrow{a}$ of magnitude 4 m/$s^2$. What angle $\theta$ between $\overrightarrow{v}$ and the positive direction of the y axis would result in a collision?

The position of a particle moving along x-axis is given by $x=3t^2-t^3$; where x is in m and t is in sec. Consider the following statements:

A solenoid of 2.5 metre length and 2.0 cm diameter possesses 10 turns per cm. A current of 0.5 ampere is flowing through it. The magnetic induction at axis inside the solenoid is: