A ball is made of a material of density $$\rho$$ where $${\rho }_{oil}<\rho <{\rho }_{water}$$ with $${\rho }_{oil}$$ and $${\rho }_{water}$$ representing the densities of oil and water, respectively. The oil and water are immiscible. If the above ball is in equilibrium in a mixture of this oil and water, which of the following pictures represents its equilibrium position?

For a particle in uniform circular motion the acceleration $$\overrightarrow{a}$$ at a point P(R, θ) on the circle of radius R is (here θ is measured from the x–axis)

A point $$P$$ moves in counter-clockwise direction on a circular path as shown in the figure. The movement of $$P$$ is such that it sweeps out a length $$s=t^3+5,$$ where $$s$$ is in metres and $$t$$ is in seconds. The radius of the path is $$20$$ $$m.$$ The acceleration of $${}^{\prime }{P}^{\prime }$$ when $$t=2$$ $$s$$ is nearly.

The combination of gates shown below yields

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by U\left( x \right) = {a \over {{x^{12}}}} - {b \over {{x^6}}}, where $$a$$ and $$b$$ are constants and $$x$$ is the distance between the atoms. If the dissociation energy of the molecule is $$D=\left[U\left(x=\infty \right)-U_{atequilibrium}\right],D$$ is

In a series $$LCR$$ circuit $$R=200\Omega$$ and the voltage and the frequency of the main supply is $$220V$$ and $$50$$ $$Hz$$ respectively. On taking out the capacitance from the circuit the current lags behind the voltage by $$30^{\circ }.$$ On taking out the inductor from the circuit the current leads the voltage by $$30^{\circ }.$$ The power dissipated in the $$LCR$$ circuit is

In the circuit shown below, the key $$K$$ is closed at $$t=0.$$ The current through the battery is

A nucleus of mass $$M+$$$$\Delta m$$ is at rest and decays into two daughter nuclei of equal mass {M \over 2} each. Speed of light is $$c.$$ The binding energy per nucleon for the parent nucleus is $$E_1$$ and that for the daughter nuclei is $$E_2.$$ Then

A radioactive nucleus (initial mass number $$A$$ and atomic number $$Z$$ emits $$3\alpha$$- particles and $$2$$ positrons. The ratio of number of neutrons to that of protons in the final nucleus will be

A nucleus of mass $$M+$$$$\Delta m$$ is at rest and decays into two daughter nuclei of equal mass {M \over 2} each. Speed of light is $$c.$$ The speed of daughter nuclei is

A diatomic ideal gas is used in a Carnot engine as the working substance. If during the adiabatic expansion part of the cycle the volume of the gas increases from $$V$$ to $$32$$ $$V$$, the efficiency of the engine is

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $$30^{\circ }$$ with each other. When suspended in a liquid of density $$0.8g$$ $$c{m}^{-3},$$ the angle remains the same. If density of the material of the sphere is $$1.6$$ $$g$$ $$c{m}^{-3},$$ the dielectric constant of the liquid is

Let there be a spherically symmetric charge distribution with charge density varying as \rho \left( r \right) = {\rho _0}\left( {{5 \over 4} - {r \over R}} \right) upto $$r=R,$$ and $$\rho \left(r\right)=0$$ for $$r>R,$$ where $$r$$ is the distance from the erigin. The electric field at a distance $$r\left(r<R\right)$$ from the origin is given by

A thin semi-circular ring of radius $$r$$ has a positive charges $$q$$ distributed uniformly over it. The net field $$\overrightarrow{E}$$ at the center $$O$$ is

A particle is moving with velocity $$\overrightarrow{v}=k\left(y\hat{i}+x\hat{j}\right)$$, where K is a constant. The general equation for its path is

If a source of power $$4kW$$ produces $$10^{20}$$ photons/second, the radiation belongs to a part of the spectrum called

Statement - $$1$$ : When ultraviolet light is incident on a photocell, its stopping potential is $$V_0$$ and the maximum kinetic energy of the photoelectrons is $$K_{\max }$$. When the ultraviolet light is replaced by $$X$$-rays, both $$V_0$$ and $$K_{\max }$$ increase. Statement - $$2$$ : Photoelectrons are emitted with speeds ranging from zero to a maximum value because of the range of frequencies present in the incident light.

An initially parallel cylindrical beam travels in a medium of refractive index $$\mu \left(I\right)={\mu }_0+{\mu }_{2}I,$$ where $${\mu }_0$$ and $${\mu }_2$$ are positive constants and $$I$$ is the intensity of the light beam. The intensity of the beam is decreasing with increasing radius. The initial shape of the wavefront of the beam is

Let $$C$$ be the capacitance of a capacitor discharging through a resistor $$R.$$ Suppose $$t_1$$ is the time taken for the energy stored in the capacitor to reduce to half its initial value and $$t_2$$ is the time taken for the charge to reduce to one-fourth its initial value. Then the ratio $$t_1/t_2$$ will be

The equation of a wave on a string of linear mass density $$0.04kg{m}^{-1}$$ is given by $y = 0.02\left( m \right)\,\sin \left[ {2\pi \left( {{t \over {0.04\left( s \right)}} - {x \over {0.50\left( m \right)}}} \right)} \right].$ The tension in the string is