An initially parallel cylindrical beam travels in a medium of refractive index where and are positive constants and *I* is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius.

As the beam enters the medium, it will

A transparent solid cylindrical rod has a refractive index of . It is surrounded by air. A light ray is incident at the mid-point of one end of the rod as shown in the figure

The incident angle θ for which the light ray grazes along the wall of the rod is

A long metallic bar is carrying heat from one of its ends to the other end under steady-state. The variation of temperature θ along the length x of the bar from its hot end is best described by which of the figures?

A charge *Q* is placed at each of the opposite corners of a square. A charge q is placed at each of the other two corners. If the net electrical force on *Q* is zero, then what will be the *Q/q* ?

Consider a rubber ball freely falling from a height *h* = 4.9 m onto a horizontal elastic plate. Assume that the duration of collision is negligible and the collision with the plate is totally elastic. Then what will be the velocity as a function of time and the height as function of time?

A motor cycle starts from rest and accelerates along a straight path at 2 m/s^{2}. At the starting point of the motor cycle there is a stationary electric siren. How far has the motor cycle gone when the driver hears the frequency of the siren at 94% of its value when the motor cycle was at rest? (Speed of sound = 330 ms^{-1}).

A particle has an initial velocity and an acceleration of . What will be its speed after 10 s?

The surface of a metal is illuminated with the light of 400 nm. The kinetic energy of the ejected photoelectrons was found to be 1.68 eV. The work function of the metal is (*hc* = 1240 eV nm)

Two moles of helium gas are taken over the cycle *ABCDA*, as shown in the P - T diagram

The net work done on the gas in the cycle *ABCDA* is

Two moles of helium gas are taken over the cycle *ABCDA*, as shown in the P - T diagram

The work done on the gas in taking it from *D to A* is

Two moles of helium gas are taken over the cycle *ABCDA*, as shown in the P - T diagram

Assuming the gas to be ideal the work done on the gas in taking it from A to B is

An inductor of inductance *L = 400 mH* and resistors of resistances *R _{1}* = 2 and

*R*= 2 are connected to a battery of emf 12 V as shown in the figure. The internal resistance of the battery is negligible. The switch

_{2}*S*is closed at

*t = 0.*What will be the potential drop across

*L*as a function of time?

In an experiment the angles are required to be measured using an instrument. 29 divisions of the main scale exactly coincide with the 30 divisions of the vernier scale. If the smallest division of the main scale is half-a-degree (= 0.5°), then the least count of the instrument will be

Two wires are made of the same material and have the same volume. However wire 1 has cross-sectional area *A* and wire 2 has cross-sectional area *3A*. If the length of wire 1 increases by âˆ†x on applying force *F*, how much force is needed to stretch wire 2 by the same amount?

The height at which the acceleration due to gravity becomes g/9 (where g = the acceleration due to gravity on the surface of the earth) in terms of R, the radius of the earth is

Three sound waves of equal amplitudes have frequencies *(u – 1)*, *u, (u + 1)*. They superpose to give beats. The number of beats produced per second will be

Image of an object approaching a convex mirror of radius of curvature 20 m along its optical axis is observed to move from 25/3 m to 50/7 m in 30 seconds. What is the speed of the object in km per hour?

The focal length of a thin biconvex lens is 20 cm. When an object is moved from a distance of 25 cm in front of it to 50 cm, the magnification of its image changes from m25 to m50. The ratio m25/m50 is

Two identical glass rods S1 and S2 (refractive index = 1.5) have one convex end of radius of curvature 10 cm. They are placed with the curved surfaces at a distance d as shown in the figure, with their axes (shown by the dashed line) aligned. When a point source of light P is placed inside rod S1 on its axis at a distance of 50 cm from the curved face, the light rays emanating from it are found to be parallel to the axis inside S2.The distance d is

A point source S is placed at the bottom of a transparent block of height 10 mm and refractive index 2.72. It is immersed in a lower refractive index liquid as shown in the figure. It is found that the light emerging from the block to the liquid forms a circular bright spot of diameter 11.54 mm on the top of the block. The refractive index of the liquid is?

A ray of light travelling in the direction is incident on a plane mirror. After reflection, it travels along the direction . The angle of incidence is

A bi-convex lens is formed with two thin plano convex lenses. Refractive index n of the first lens is 1.5 and that of the second lens is 1.2. Both the curved surfaces are of the same radius of curvature R = 14 cm. For this bi-convex lens, for an object distance of 40 cm, the image distance will be.

The image of an object, formed by a plano-convex lens at a distance of 8 m behind the lens, is real is one-third the size of the object. The wavelength of light inside the lens is 3 times the wavelength in free space.The radius of the curved surface of the lens is

A biconvex lens of focal length 15 cm is in front of a plane mirror. The distance between the lens and the mirror is 10 cm. A small object is kept at a distance of 30 cm from the lens. The final image is

A ball is dropped from a height of 20 m above the surface of water in a lake. The refractive index of water is 4/3. A fish inside the lake, in the line of fall of the ball, is looking at the ball. At an instant, when the ball is 12.8 m above the water surface, the fish sees the speed of ball as [Take g = 10 m/s2.]

A ray of light traveling in water is incident on its surface open to air. The angle of incidence is , which is less than the critical angle. Then there will be

In an experiment to determine the focal length (f) of a concave mirror by the u-v method, a student places the object pin A on the principal axis at a distance x from the pole P. The student looks at the pin and its inverted image from a distance keeping his/her eye in line with PA. When the student shifts his/her eye towards left, the image appears to the right of the object pin. Then

A thin lens has a focal length f and its aperture has a diameter d. It forms an image of intensity I. Now, the outer part extending from R/2 to R is blackened. The focal length of the lens and intensity of image becomes/remain (R = radius of aperture = d/2 )

The total mechanical energy of the particle is

A body of mass m is slowly halved up the rough hill by a force F at which each point is directed along a tangent to the hill.

work done by the force

The upper half of an inclined plane with inclination is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by

A particle is projected along the inner surface of a smooth vertical circle of radius R, its velocity at the lowest point

being

It will leave the circle at an angular distance.... from the highest point

A pendulum bob of mass m is suspended at rest. A cosntant horiozntal force F = mg/2 starts acting on it. The maximum angular deflection of the string is

Consider a roller coaster with a circular loop. A roller coaster car starts from rest from the top of a hill which is 5 m higher than the top of the loop. It rolls down the hill and through the loop. What must the radius of the loop be so that the passengers of the car will feel at highest point, as if they have their normal weight?

One end of a light rope is tied directly to the ceiling. A man of mass M initially at rest on the ground starts climbing the rope hand over hand upto a height l. From the time he starts at rest on the ground to the time he is hanging at rest at a height l, how much work was done on the man by the rope?

A car is moving along a hilly road as shown (side view). The coefficient of static friction between the tyres and pavement is constant and the car maintains a steady speed. If, at one of the points shown the driver applies the brakes as hard as possible without making the tyres slip, the magnitude of the frictional force immediately after the brakes are applied will be maximum if the car was at

A box of mass m is initially at rest on a horizontal surface. A constant horizontal force of mg/2 is applied to the box directed to the right. The coefficient of friction of the surface changes with the distance pushed as where x is the distance from the initial location. For what distance is the box pushed until it comes to rest again?

When a conservative force does positive work on a body, then

A solid sphere of mass M and radius R is surrounded by a spherical shell of same mass M and radius 2R as shown. A small particle of mass m is released from rest from a height h (<<R) above the shell. There is a hole in the shell.

What time will it take to move from A to B?

A solid sphere of mass M and radius R is surrounded by a spherical shell of same mass M and radius 2R as shown. A small particle of mass m is released from rest from a height h (<<R) above the shell. There is a hole in the shell.

In what time will it enter the hole at A :–

If the law of gravitation be such that the force of attraction between two particles vary inversely as the 5/2th power of their separation, then the graph of orbital velocity v0 plotted against the distance r of a satellite from the earth's centre on a log-log scale is shown alongside. The slope of line will be

A particle is projected from point A, that is at a distance 4R from the centre of the Earth, with speed v1 in a direction making 30 with the line joining the centre of the Earth and point A, as shown. Find the speed v1 of particle (in m/s) if particle passes grazing the surface of the earth. Consider gravitational interaction only between these two.

Gravitational potential difference between a point on surface of planet and another point 10 m above is 4 J/kg. Considering gravitational field to be uniform, how much work is done in moving a mass of 2 kg from the surface to a point 5m above the surface?

Calculate the mass of the sun if the mean radius of the earth's orbit is and

A mass ( = mass of earth) is to be compressed in a sphere in such a way that the escape velocity from its surface is (equal to that of light). What should be the radius of the sphere?

A particle is projected vertically upwards the surface of the earth (radius Re) with a speed equal to one fourth of escape velocity. What is the maximum height attained by it from the surface of the earth ?

The escape velocity for a planet is Ve. A particle starts from rest at a large distance from the planet, reaches the planet only under gravitational attraction, and passes through a smooth tunnel through its centre. Its speed at the centre of the planet will be

Two particles of equal mass m go round a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is

Two moles of ideal helium gas are in a rubber balloon at 30oC. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to 35oC. The amount of heat required in raising the temperature is nearly (take R = 8.31 J/mol.K)

An ideal gas is expanding such that PT*T = constant. The coefficient of volume expansion of the gas is