A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1 m and its crosssectional area is 4.9 $$ \times $$ 10^-7 m². If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency 140 rad s⁻¹. If the Young’s modulus of the material of the wire is n $$ \times $$ 10⁹ Nm^-2, the value of n is

A block of mass m is on an inclined plane of angle θ. The coefficient of friction between the block and the plane is μ and tan θ > μ. The block is held stationary by applying a force P parallel to the plane. The direction of force pointing up the plane is taken to be positive. As P is varied from P₁ = mg(sinθ − μ cosθ) to P₂ = mg(sinθ + μ cosθ), the frictional force f versus P graph will look like

A point mass of 1 kg collides elastically with a stationary point mass of 5 kg. After their collision, the 1 kg mass reverses its direction and moves with a speed of 2 ms⁻¹. Which of the following statement(s) is (are) correct for the system of these two masses?

A binary star consists of two stars A (mass 2.2M_s) and B (mass 11M_s), where M_s is the mass of the sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is

Gravitational acceleration on the surface of a planet is $${{\sqrt 6 } \over {11}}g$$, where $$g$$ is the gravitational acceleration on the surface of the earth. The average mass density of the planet is $${2 \over 3}$$ times that of the earth. If the escape speed on the surface of the earth is taken to be 11 kms^-1, the escape speed on the surface of the planet in kms^-1 will be

A real gas behaves like an ideal gas if its

A piece of ice (heat capacity = 2100 J kg^{-1 o}C^-1 and latent heat = 3.36 $$ \times $$ 10⁵ J kg^-1 ) of mass m grams is at - 5 ^oC at atmospheric pressure. It is given 420 J of heat so that the ice starts melting. Finally when the ice-water mixture is in equilibrium, it is found that 1 gm of ice has melted. Assuming there is no other heat exchange in the process, the value of m is

Two spherical bodies A (radius 6 cm ) and B (radius 18 cm ) are at temperature T₁ and T₂, respectively. The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm. Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to that of B?

A few electric field lines for a system of two charges $${Q_1}$$ and $${Q_2}$$ fixed at two different points on the $$x$$-axis are shown in the figure. These lines suggest that

A student uses a simple pendulum of exactly 1m length to determine g, the acceleration due to gravity. He uses a stop watch with the least count of 1 sec for this and records 40 seconds for 20 oscillations. For this observation, which of the following statement(s) is (are) true?

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, 100, 60 and 40 W bulbs have filament resistances R₁₀₀, R₆₀ and R₄₀ respectively, the relation between these resistances is

To verify Ohm's law, a student is provided with a test resitor R_T, a high resistance R₁, a small resistance R₂, two identical galvanometers G₁ and G₂, and a variable voltage source V. The correct circuit to carry out the experiment is

An AC voltage source of variable angular frequency $$\omega$$ and fixed amplitude V₀ is connected in series with a capacitance C and an electric bulb of resistance R (inductance zero). When $$\omega$$ is increased

A thin flexible wire of length L is connected to two adjacent fixed points and carries a current I in the clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength B going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is

A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Consider a thin square sheet of side L and thickness, made of a material of resistivity $$\rho$$. The resistance between two opposite faces, shown by the shaded areas in the figure is

One mole of an ideal gas in initial state A undergoes a cyclic process ABCA, as shown in the figure. Its pressure at A is P₀. Choose the correct option(s) from the following:

A ray OP of monochromatic light is incident on the face AB of prism ABCD near vertex B at an incident angle of 60$$^\circ$$ (see figure). If the refractive index of the material of the prism is $$\sqrt3$$, which of the following is(are) correct?

In the graph below, the resistance R of a superconductor is shown as a friction of its temperature T for two different magnetic fields B₁ (solid line) and B₂ (dashed line). If B₂ is larger than B₁ which of the following graphs shows the correct variation of R with T in these fields?

A superconductor has T_c(0) = 100 K. When a magnetic field of 7.5 T is applied, its T_c decreases to 75 K. For this material, one can definitely say that when

If the total energy of the particle is E, it will perform periodic motion only if

For periodic motion of small amplitude A, the time period T of this particle is proportional to

The acceleration of this particle for $$|x| > {X_0}$$ is

A stationary source is emitting sound at a fixed frequency f₀, which is reflected by two cars approaching the source. The difference between the frequencies of sound reflected from the cars is 1.2% of f₀. What is the difference in the speeds of the cars (in km per hour) to the nearest integer? The cars are moving at constant speeds much smaller than the speed of sound which is 330 ms^$$-$$1.

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 $${{{m_{25}}} \over {{m_{50}}}}$$ is __________.

An $$\alpha$$-particle and a proton are accelerated from the rest by a potential difference of 100 V. After this, their de Broglie wavelengths are $$\lambda$$_$$\alpha$$ and $$\lambda$$_p, respectively. The ratio $${{{\lambda _p}} \over {{\lambda _\alpha }}}$$, to the nearest integer, is _____________.

When two identical batteries of internal resistance 1 $$\Omega$$ each are connected in series across a resistor R, the rate of heat produced in R is J₁. When the same batteries are connected in parallel across R, the rate is J₂. If J₁ = 2.25 J₂, then the value of R in $$\Omega$$ is __________.

When two progressive waves $${y_1} = 4\sin (2x - 6t)$$ and $${y_2} = 3\sin \left( {2x - 6t - {\pi \over 2}} \right)$$ are superimposed, the amplitude of the resultant wave is __________.