A planet having mass \(9\,Me\) and radius \(4 R _e\), where Me and \(\operatorname{Re}\) are mass and radius of earth respectively, has escape velocity in \(km / s\) given by:(Given escape velocity on earth \(\left.V_e=11.2 \times 10^3\,m / s \right)\)

A piston of mass \(M\) is hung from a massless spring whose restoring force law goes as \(\mathrm{F}=-\mathrm{kx}^3\), where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' \(n\) ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height \(\mathrm{L}_0\) to \(\mathrm{L}_1\), the total energy delivered by the filament is (Assume spring to be in its natural length before heating)

Young's moduli of the material of wires \(A\) and \(B\) are in the ratio of \(1: 4\), while its area of cross sections are in the ratio of \(1: 3\). If the same amount of load is applied to both the wires, the amount of elongation produced in the wires \(A\) and \(B\) will be in the ratio of [Assume length of wires \(A\) and \(B\) are same]

An electron, a doubly ionized helium ion \(\left( He ^{++}\right)\) and a proton are having the same kinetic energy. The relation between their respective de\(-\)Broglie wavelengths \(\lambda_{ e }, \lambda_{ He ^{++}}\) and \(\lambda_{ P }\) is

A sample of gas at temperature \(\mathrm{T}\) is adiabatically expanded to double its volume. Adiabatic constant for the gas is \(\gamma=3 / 2\). The work done by the gas in the process is _______. \((\mu=1 \mathrm{~mole})\)

A monochromatic light wave with wavelength \(\lambda_1\) and frequency \(v_1\) in air enters another medium. If the angle of incidence and angle of refraction at the interface are \(45^{\circ}\) and \(30^{\circ}\) respectively, then the wavelength \(\lambda_2\) and frequency \(v_2\) of the refracted wave are :

Two stars of masses \(3\times10^{31}\, kg\) each, and at distance \(2\times10^{11}\, m\) rotate in a plane about their common centre of mass \(O. A\) meteorite passes through \(O\) moving perpendicular to the star’s rotation plane. In order to escape from the gravitational field of this double star, the minimum speed that meteorite should have at \(O\) is : (Take Gravitational constant \(G = 6.67\times10^{-11}\, Nm^2\, kg^{-2}\))

In a hydrogen like atom electron make transition from an energy level with quantum number \(n\) to another with quantum number \((n- 1)\). If \(n > >  1\), the frequency ofradiation emitted is proportional to

A vessel of depth \(d\) is half filled with oil of refractive index \(n_1\) and the other half is filled with water of refractive index \(n_2\). The apparent depth of this vessel when viewed from above will be

In Young's double slits experiment, the position of \(5^{\text {th }}\) bright fringe from the central maximum is \(5\,cm\). The distance between slits and screen is \(1\,m\) and wavelength of used monochromatic light is \(600\,nm\). The separation between the slits is \(........\mu m\)