A uniform circular disc of mass 50 kg and radius 0.4 m is rotating with an angular velocity of 10 rad s^-1 about its own axis, which is vertical. Two uniform circular rings, each of mass 6.25 kg and radius 0.2 m, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity (in rad s^-1 ) of the system is

The image of an object, formed by a plano-convex lens at a distance of 8 m behind the lens, is real and is one-third the size of the object. The wavelength of light inside the lens is $${2 \over 3}$$ times the wavelength in free space. The radius of the curved surface of the lens is

The work done on a particle of mass m by a force $$$K\left[ {{x \over {{{\left( {{x^2} + {y^2}} \right)}^{3/2}}}}\widehat i + {y \over {{{\left( {{x^2} + {y^2}} \right)}^{3/2}}}}\widehat j} \right]$$$ (K being a constant of appropriate dimensions), when the particle is taken from the point $$\left( {a,0} \right)$$ to the point $$\left( {0,a} \right)$$ along a circular path of radius a about the origin in the x-y plane is

A particle of mass 0.2 kg is moving in one dimension under a force that delivers a constant power 0.5 W to the particle. If the initial speed (in m/s) of the particle is zero, the speed (in m/s) after 5 s is

A particle of mass m is projected from the ground with an initial speed u₀ at an angle $$\alpha $$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed u₀. The angle that the composite system makes with the horizontal immediately after the collision is

A bob of mass m, suspended by a string of length l₁ is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length l₂, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $${{{l_1}} \over {{l_2}}}$$ is

One end of a horizontal thick copper wire of length 2L and radius 2R is welded to an end of another horizontal thin copper wire of length L and radius R. When the arrangement is stretched by applying forces at two ends, the ratio of the elongation in the thin wire to that in the thick wire is

A solid sphere of radius R and density $$\rho $$ is attached to one end of a mass-less spring of force constant k. The other end of the spring is connected to another solid sphere of radius R and density 3$$\rho $$. The complete arrangement is placed in a liquid of density 2$$\rho $$ and is allowed to reach equilibrium. The correct statement(s) is (are)

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is

Two non-conducting solid spheres of radii $$R$$ and $$2R,$$ having uniform volume charge densities $${\rho _1}$$ and $${\rho _2}$$ respectively, touch each other. The net electric field at a distance $$2$$ $$R$$ from the center of the smaller sphere, along the line joining the centers of the spheres, is zero. The ratio $${{{\rho _1}} \over {{\rho _2}}}$$ can be

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance $$C.$$ The switch $${S_1}$$ is pressed first to fully charge the capacitor $${C_1}$$ and then released. The switch $${S_2}$$ is then pressed to charge the capacitor $${C_2}.$$ After some time, $${S_2}$$ is released and then $${S_3}$$ is pressed. After some time

The diameter of a cylinder is measured using a Vernier callipers with no zero error. It is found that the zero of the Vernier scale lies between 5.10 cm and 5.15 cm of the main scale. The Vernier scale has 50 divisions equivalent to 2.45 cm. The 24^th division of the Vernier scale exactly coincides with one of the main scale divisions. The diameter of the cylinder is

A ray of light travelling in the direction $${1 \over 2}\left( {\widehat i + \sqrt 3 \widehat j} \right)$$ is incident on a plane mirror. After reflection, it travels along the direction $${1 \over 2}\left( {\widehat i - \sqrt 3 \widehat j} \right)$$. The angle of incidence is

Two rectangular blocks, having identical dimensions, can be arranged in either configuration-I or configuration-II as shown in the figure. One of the blocks has thermal conductivity $$\kappa $$ and the other 2$$\kappa $$. The temperature difference between the ends along the x-axis is the same in both the configurations. It takes 9 s to transport a certain amount of heat from the hot end to the cold end in configuration-I. The time to transport the same amount of heat in configuration-II is

A pulse of light of duration 100 ns is absorbed completely by a small object initially at rest. Power of the pulse is 30 mW and the speed of light is 3 $$\times$$ 10⁸ ms^$$-$$1. The final momentum of the object is

In the Young's double-slit experiment using a monochromatic light of wavelength $$\lambda$$, the path difference (in terms of an integer n) corresponding to any point having half the peak intensity is

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation

y(x, t) = (0.01 m) sin[(62.8 m^$$-$$1)x] cos[(628 s^$$-$$1)t]

Assuming $$\pi$$ = 3.14, the correct statement(s) is(are)

A particle of mass M and positive charge Q, moving with a constant velocity $${\overrightarrow u _1} = 4\widehat i$$ ms^$$-$$1 enters a region of uniform static magnetic field, normal to the xy plane. The region of the magnetic field extends from x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side after 10 ms with a velocity $${\overrightarrow u _2} = 2\left( {\sqrt 3 \widehat i + \widehat j} \right)$$ ms^$$-$$1. The correct statement(s) is(are)

The work functions of silver and sodium are 4.6 and 2.3 eV, respectively. The ratio of the slope of the stopping potential versus frequency plot for silver to that of sodium is ___________.

A freshly prepared sample of a radioisotope of half-life 1386 s has activity 10³ disintegrations per second. Given that ln2 = 0.693, the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first 80 s after preparation of the sample is __________.