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A sphere has only translational velocity $V_0=2r\omega$ when placed on rough horizontal surface with frictional coefficient m. If r is radius then the time in which pure rolling starts is

At a certain point on a screen in a YDSE apparatus, the path difference for the two interfering waves is ${\left(\frac{1}{3}\right)}^{rd}$ of the wavelength. The ratio of the intensity at this point to that at the centre of a bright fringe is

A coil carrying a heavy current and having large number of turns is mounted in a N-S vertical plane and a current flows in clockwise direction as seen by a person on the east. A small magnetic needle at its centre will have its north pole in

Find the force exerted by a light beam of intensity I, incident on a cylinder (height h and base radius R) placed on a smooth surface as shown in figure if surface of cylinder is having reflection coefficient 0.8. (assume no transmission)

In the circuit shown below, all the inductors (assumed ideal) and resistors are identical. The current through the resistance on the right is I after the key K has been switched on for a long time. The currents through the three resistors(in order, from left to right) immediately after the key is switched off are

If the mass of a body is M on the surface of the earth, the mass of the same body on the surface of the moon is:

The radius of gyration of uniform disc of radius $r$, about an axis passing through $r/4$ from center of disc and perpendicular to plane of disc is

A sphere is projected up on an inclined plane with a velocity $v_o$ and angular velocity as shown. The coefficient of friction between the sphere and the plane is $\mu =tan\theta$. Find the total time of rise of the sphere.

The area of the plates of a parallel plate capacitor is A and the gap between them is d. The gap is filled with a non - homogeneous dielectric whose dielectric constant varies with the distance y from one plate as $k=\lambda sec\left(\frac{\pi y}{2d}\right)$, where $\lambda$ is a dimensionless constant. The capacitance of this capacitor is

A uniform chain of length $l$ and mass $m$ lies on a smooth table. A very small part of this chain hangs from the table. It begins to fall under the weight of the hanging end. Find the velocity of the chain when the length of the hanging part becomes $y$.

Two particles $A$ and $B$ moving in x-y plane are at origin at $t=0$ second. The initial velocity vectors of $A$ and $B$ are $\overrightarrow{u_A}=8\hat{i}\frac{m}{s}\text{and}\overrightarrow{u_B}=8\hat{j}\frac{m}{s}$. The acceleration of $A$ and $B$ are constant and are $\overrightarrow{a_A}=-2\hat{i}\frac{m}{s^2}\text{and}\overrightarrow{a_B}=-2\hat{j}\frac{m}{s^2}$.
Column I gives certain statements regarding particles $A$ and $B$. Column II gives corresponding results. Match the statements in column I with corresponding results in Column II.
$\begin{array}{cc}\text{Column-I}& \text{Column-II}\\ \text{(A)The time (in seconds) at which velocity}& \left(p\right)16\sqrt{2}\\ \text{of A relative to B is zero}& \\ \text{(B)The distance (in metres) between A and B}& \left(q\right)8\sqrt{2}\\ \text{when their relative velocity is zero}& \\ \text{(C)The time (in seconds) after t = 0 second}& \left(r\right)8\\ \text{at which A and B are at same position}& \\ \text{(D)The magnitude of relative velocity(in}m{s}^{-1}\text{) of A and B}& \left(s\right)4\\ \text{at the instant they are at same position}& \end{array}$

Two blocks of masses $m_1$ and $m_2$ interconnected by a spring of stiffness $k$ are placed on a horizontal surface. If a constant horizontal force $F$ acts on the block $m_1$ it slides through a distance $x$ whereas $m_2$ remains stationary. If the coefficient of friction between all contacting surfaces is $\mu$, find the speed of the block $m_1$ as a function of $x$.

An ideal gas with adiabatic exponent $\gamma$ is heated at a constant pressure. It absorbs total $Q$ amount of heat. Fraction of the heat used to increase its temperature of the gas is

For the circuit shown in figure, initially key $K_1$ is closed and key $K_2$ is kept open for a long time. Then $K_1$ was opened and $K_2$ was closed, what will be the charge on capacitors $C_2\&C_3$ finally? $[C=1\mu F]$

A rod is placed inside a hollow spherical shell as shown in figure. Friction acts between the rod and the shell. The number of forces to be shown in FBD of rod is/are?

What is the moment of inertia in a thin hollow sphere touching the edge (tangent)???

A famous relation in physics relates moving mass m to the rest mass $m_0$ of a particle in terms of its

speed v and the speed of light c. (This relation first arose as a consequence of special theory of relatively

given by Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant

c. He writes $m=\frac{m_0}{(1-v^2{)}^{\frac{1}{2}}}.$ Guess where to put the missing c?

The area of cross section of the wider tube shown in figure is 900 $c{m}^2$. If the boy standing on the piston weighs 45 kg, find the difference in the levels of water in the two tubes.

A particle is dropped under gravity from rest from a height h(g=9.8) and it travels a distance 9h/25 in the last second,the height h is

A block is moving on an inclined plane making an angle ${45}^{\circ }$ with the horizontal and the coefficient of friction is $\mu$. The force required to just push it up the inclined plane is $3$ times the force required to just prevent it from sliding down. If we define $n=10\mu$, then $n$ is

Ten solid spheres of same radius having densities $\rho ,2\rho ,3\rho$... $10\rho$ respectively are put on the $x$ - axis at positions $x=1,x=2,x=3$ ... $x=10$. Find the position of centre of mass of these ten spheres.

A prticle of mass $m$ moves with velocity $v_0=20\text{m/s}$ towards a large wall that is moving with velocity $v=5\text{m/s}$ towards the particle as shown. If the particle collides with the wall elastically, then find the speed of the particle just after collision. (Assume collision with the wall is elastic)

The value of electric potential throughout the volume of a conductor is

A Muon is a particle that has the same charge as an electron but is 200 times heavier than it . if we had an atoms in which the muon revolves around a Proton instead of an electron what would be the value of magnetic moment of the muon in the ground state of such an atom

In the circuit shown in Fig, capacitors $A$ and $B$ have identical geometry, but a material of dielectric constant 3 is present between the plates of $B$. The potential difference across $A$ and $B$ are respectively

Prove that the path of a projectile with respect to another projectile is a straight line?

Unpolarized light of intensity $32\frac{W}{m^2}$ passes through three polarizers such that the transmission axis of the last polarizer is crossed with that of the first. The intensity of final emerging light is $3\frac{W}{m^2}$ . The angle between the first two transmission axes of the polarizers is:

A motorboat travelling at $10m/s$ due North, encounters a current travelling at $5m/s$ due East. Find the magnitude of relative velocity of boat with respect to the current.

For a simple pendulum, the graph between g and $T^2$ will be

Which of the following is not true about Bohr's model of the atom?

A particle describes a circular motion in horizontal plane in a conical funnel with speed v, the height of plane of circle from bottom is 40 cm, find v.

A planet has mass $\frac{1}{10}$ of that of earth, while radius is $\frac{1}{3}$ that of earth. If a person can throw a stone on earth surface to a height of 90m, then he will be able to throw the stone on that planet to a height

In a model experiment as shown in the figure there is a fixed inclined plane.
A thin ring of radius ‘r’ is rolling upwards along the incline without slipping at a velocity $v_0$ from the bottom. In three different cases of the experiment, everything is kept same except the coefficient of friction of the inclined surface. It is changed as $\mu =0,\mu =\tan \theta \&\mu =2\tan \theta$ respectively. In each case the height to which the ring rises on the incline before stopping momentarily is $\Delta h_1,\Delta h_2,\Delta h_3$ respectively. Then

Soap helps in cleaning clothes, because

A 10 -cm long rod carries a charge of $+50\mu C$ distributed uniformly along its length. Find the magnitude of the electric field at a point 10 cm from from both the ends of the rod.

When a pressure of 100 atmosphere is applied on a spherical ball of rubber, then its volume reduces to $0.01\%$. The bulk modulus of the material of the rubber in $\frac{dyne}{c{m}^2}$ is

A bar magnet is hung by a thin cotton thread in a uniform horizontal magnetic field and is in equilibrium state. The energy required to rotate it by $60°$ is $W$. Now the torque required to keep the magnet in this new position is

The equation for the vibration of a string, fixed at both ends vibrating in its third harmonic, is given

$y=\left(0.4cm\right)sin\left[\right(0.314c{m}^{-1}\left)x\right]cos\left[\right(600\pi s^{-1}\left)t\right]$

i. What is the frequency of vibration?

ii. What are the positions of the nodes?

iii. What is the length of the string?

iv. What is the wavelength and the speed of two travelling waves that can interfere to give this vibration?

A bock of mass $0.9\text{kg}$ is attached to a spring of force constant k is lying on a frictionless floor. The spring is compressed to $\sqrt{2}\text{cm}$ and the block is at a distance of $\frac{1}{\sqrt{2}}\text{cm}$ from the wall as shown in the figure. When the block is released, it makes an elastic collision with the wall and its period of motion is $0.2\text{s}$. Find the approximate value of $k$.

A particle is performing a linear simple harmonic motion. If the acceleration and the corresponding velocity of the particle are a and v respectively, then which of the following graphs is correct?

If velocity of a particle is $v\left(t\right)=(15-3t)\text{m/s}$, the average speed of the particle in $6\text{sec}$ is

A steel wire is suspended vertically from a rigid support. When loaded with a weight in air, it expands by La and when the weight is immersed completely in water, the extension is reduced to $L_w$. Then relative density of the material of the weight is

The frequency of the fourth harmonic of an organ pipe is $33\text{Hz}$ less than the frequency of the fifth harmonic of the same organ pipe. Find the difference in wavelengths for the given two cases. Neglect end corrections.
[speed of sound in air $=330\text{m/s}$]