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An electromagnetic wave of v = 3MHz passes from vacuum into dielectric medium with $ϵ=4.0{ϵ}_0.$ Then:

A particle of mass m is attached to a light string of length l, the other end of which is fixed. Initially the string is kept horizontal and the particle is given an upward velocity v. The particle is just able to complete a circle.

A man who is running has half the kinetic energy of the boy of half his mass. The man speeds up by $1m/s$ and then has the same kinetic energy as the boy. The original speed of the man was

Weight of a body of mass m decreases by $1\%$ when it is raised to height h above the earth’s surface. If the body is taken to a depth h in a mine, change in its weight is

Two persons A and B having masses $55\text{kg}$ & $65\text{kg}$ respectively, are sitting at the opposite ends of a stationary boat. The length of the boat is $3\text{m}$ and the weight is $100\text{kg}$. Person A walks upto person B & sits with him. If the boat is in still water, the centre of mass of the system shifts by

Two cars $A$ and $B$ are travelling in the same direction with velocities $v_1$ and $v_2$ $(v_1>v_2)$. When the car $B$ is at a distance $d$ ahead of the car $A$, the driver of the car $A$ applied the brake producing a uniform retardation $a$. There will be no collision if

A ball of mass $5\text{kg}$ moving at speed $2\text{m/s}$ makes a head on collision with an identical ball at rest. The velocities of the balls after the perfectly elastic collision between them are respectively:-

If a particle is moving clockwise in a circle in $x-y$ plane with centre at origin, the direction of angular velocity of the particle is

Two bodies of mass $4\text{kg}$ and $6\text{kg}$ are attached to the ends of a string passing over a pulley as shown in the figure below. The $4\text{kg}$ mass is attached to the tabletop by another string. The tension in this string $T_1$ is equal to (take $g=10\frac{\text{m}}{{\text{s}}^2}$ )

Specific heat capacity at constant volume of one mole diatomic ideal gas is

A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motions, find the phase difference between the individual motions.

A particle performing simple harmonic motion undergoes displacement of $\frac{A}{2}$ (where $A$ is the amplitude of simple harmonic motion) during first second. At $t=0$, the particle may be at the extreme position or mean position. The period of the simple harmonic motion can be

The height of water level in a tank is $H=100\text{cm}$. The water stream is coming out of a hole at the depth of $20\text{cm}$ from top surface of water. Find out the time taken by the water coming out of the hole to reach the ground and the horizontal range of water. (Take $g=1000{\text{cm/s}}^2$)

The figure shows a conducting ring of radius $R$. A uniform steady magnetic field $B$ lies perpendicular to the plane of the ring in a circular region of radius $r(<R)$. If the resistance per unit length of the ring is $\lambda$, then the current induced in the ring when its radius gets doubled is

A rigid body of mass $m$ and radius $R$ rolls without slipping on a rough surface. A force is acting on the rigid body at distance $x$ from the centre as shown in figure. Find the value of $x$ so that static friction will be zero.

A simple pendulum is suspended from the ceiling of a car accelerating uniformly on a horizontal road. If the acceleration is $a_0$ and the length of the pendulum is l, find the time period of small oscillations about the mean position.

The resultant of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ subtends an angle of ${45}^{\circ }$ with either of them. The magnitude of the resultant is

Solve the integral I = ${\int }_0^{\pi }si{n}^{2}xdx$

Find the MOI of four identical solid spheres each of mass $M$ and radius $R$ placed in a horizontal plane as shown in figure. Line $X{X}^{\prime }$ touches two spheres and passes through the diameter of the other two spheres.

Minute hand of a watch is $4\text{cm}$ long. How far does it's tip moves in $45\text{minutes}$? (use $\pi =3.14$)

A tape measures the length of a copper rod as $90\text{cm}$ at ${10}^{\circ }C$. What would the tape read for the length of the rod at ${30}^{\circ }C$?

Given, ${\alpha }_{\text{Cu}}=1.7\times {{10}^{-5}}^{\circ }{C}^{-1}$

A ring of mass $m$ and radius $R$ rotates about an axis passing through its centre and perpendicular to its plane with angular velocity $\omega$. Find the velocity of transverse pulse in the ring.

Out of gravitational, electromagnetic, Vander Waals, electrostatic and nuclear forces; which two are able to provide an attractive force between two neutrons

One mole of an ideal monatomic gas has initial temperature $T_0$ is made to go through the cycle $abca$ as shown in the given figure. If $U$ denotes the internal energy, then choose the correct alternative.

Three rods of identical cross-sectional area and made from the same metal form the sides of an isosceles triangle ABC, right angled at B. The points A and B are maintained at $T$ and $\sqrt{2}T$ respectively. In the steady state, the temperature of the point C is $T_C$. Assuming that only heat conduction takes place, $\frac{T_C}{T}$ is

The initial velocity of a particle is $10\text{m/s}$ and its retardation is $2{\text{m/s}}^2$. The distance moved by the particle in $5^{\text{th}}$ second of its motion is:

If the range of a gun which fires a shell with muzzle speed V is R , then the angle of elevation of the gun is

Four particles, each of mass m, are placed at the corners of a square of side a as shown in the figure. The position vector of the centre of mass is

The unit vector perpendicular to both $\hat{i}+\hat{j}$ and $\hat{j}-\hat{k}$ is

Calculate the stress developed inside a tooth cavity filled with copper when hot tea at temperature of ${57}^{\circ }\text{C}$ is drunk. You can take body (tooth) temperature to be ${37}^{\circ }\text{C}$ and ${\alpha }_{Cu}=1.7\times {10}^{-5}{/}^{\circ }\text{C}$, bulk modulus for copper $K_{Cu}=140\times {10}^9{\text{N/m}}^2$

The position of point $B$ with respect to point $A$ is

Two balls of masses $m_1=1\text{kg}$ and $m_2=2\text{kg}$ are attached to a massless rod of length $(l=1\text{m})$. Two parallel forces having equal magnitude of $F=1\text{N}$, opposite in direction (couple) are acting on both the balls as shown in the figure. Find the angular acceleration of the system ($\alpha$) about an axis passing through mid-point of the rod and also perpendicular to the plane of the rod. (Take anti-clockwise rotation as +ve)

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

A string of length $1\text{m}$ is fixed at one end with a bob of mass $100\text{g}$ and the string makes $\left(\frac{2}{\pi }\right){\text{rev s}}^{-1}$ around a vertical axis through a fixed point. The angle of inclination of the string with the vertical is

A long cylindrical conductor of radius 'a' has two cylindrical cavities each of diameter 'a' through its entire length as shown in the figure. A current I is directed out of the page and is uniform throughout the cross-section of the conductor. The magnetic field at point $P_1$ is

A truck starts from rest with an acceleration of $1.5{\text{m/s}}^2$ while a car $150\text{m}$ behind starts from rest with an acceleration of $2{\text{m/s}}^2$. What is the distance travelled by the car before it gets side by side to the truck?

Particle $P$ shown in figure is moving in a circle of radius $R=10cm$ with linear speed $v=2m/s$. Find the angular speed of particle about point $O$

The lengths of steel and copper rods are so that the length of the steel rod is 5 cm longer than that of the copper rod at all temperatures, then length of each rod, are ($\alpha$ for copper = $1.7\times {10}^{-5}{/}^{\circ }C$ and $\alpha$ for steel = $1.1\times {10}^{-5}{/}^{\circ }C$) -

If g is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass m raised from the surface of earth to a height equal to the radius of the earth R, is

A particle has initial velocity as $u_x$ i + $u_y$j and acceleration as -$a_y$. Find displacement in x when the displacement in Y becomes 0?

Two Carnot engines $A$ and $B$ are operated in series. The first one $A$ receives heat from a source at temperature $800\text{K}$ and rejects heat to a reservoir at temperature $T\text{K}$. The second engine $B$ receives the heat rejected by the first engine and in turn rejects to a heat reservoir at temperature $300\text{K}$ Find the value of temperature $T$ if the efficiencies of two engines are equal.

(Take $\sqrt{6}=2.45$ for calculation)

On a $P-T$ diagram, a cyclic process is performed as shown. Where is the volume maximum?

A pulse (shown here) is reflected from the rigid wall $A$ and then from free end $B$. The shape of the string after these $2$ reflections will be -

In an interference experiment, third bright fringe is obtained at a point on the screen with a light of $700\text{nm}$. What should be the wavelength of the light in order to obtain $5^{th}$ bright fringe at the same point?

A particle is projected vertically upward from the surface of the earth with a speed of $\sqrt{\frac{3}{2}gR}$, $R$ being the radius of the earth and g is the acceleration due to gravity on the surface of the earth. Then the maximum height ascended is (neglect cosmic dust resistance)

Find the value of force $F$, if the tension in the string connecting the $2kg$ block and $3kg$ block is $T=20N$. Consider all surfaces to be smooth.

Two stars both of mass are orbiting their common $COM$. If the distance between them is d , tangential velocity of both the stars are

The focal length of a concave mirror is f and the distance from the object to the principle focus is x. The ratio of the size of the image to the size of the object is

[Kerala PET 2005]