In a physical balance working on the principle of moments, when \(5\, mg\) weight is placed on the left pan, the beam becomes horizontal. Both the empty pans of the balance are of equal mass. Which of the following statements is correct?

A water spray gun is attached to a hose of cross sectional area \(30\) cm\(^2\). The gun comprises of \(10\) perforations each of cross sectional area \(15\) mm\(^2\). If the water flows in the hose with the speed of \(50\) cm/s, calculate the speed at which the water flows out from each perforation. (Neglect any edge effects)

A hydrogen atom, initially in the ground state is excited by absorbing a photon of wavelength \(980\,\mathop A\limits^o \). The radius of the atom in the excited state, in  terms of Bohr radius \(a_0,\) will be \((hc\,=\,12500\,eV-\mathop A\limits^o)\)

The electric field in an electromagnetic wave is given by \({E}=\left(50\, {NC}^{-1}\right) \sin \omega({t}-{x} / {c})\) The energy contained in a cylinder of volume \({V}\) is \(5.5 \times 10^{-12} \, {J}\). The value of \({V}\) is \(......{cm}^{3}\) \(\left(\right.\) given \(\left.\in_{0}=8.8 \times 10^{-12} \,{C}^{2} {N}^{-1} {m}^{-2}\right)\)

A cube is placed inside an electric field, \(\overrightarrow{{E}}=150\, {y}^{2}\, \hat{{j}}\). The side of the cube is \(0.5 \,{m}\) and is placed in the field as shown in the given figure. The charge inside the cube is \(.....\times 10^{-11} {C}\)

The decay of a proton to neutron is :

Consider a badminton racket with length scales as shown in the figure.If the mass of the linear and circular portions of the badminton racket are same \((M)\) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, \(\frac{r}{2}\) distance from the end \(A\) of the handle will be ....... \(Mr^2\)?

A ball of mass \(m\) is thrown vertically upward. Another ball of mass \(2\,m\) is thrown an angle \(\theta\) with the vertical. Both the balls stay in air for the same period of time. The ratio of the heights attained by the two balls respectively is \(\frac{1}{x}\). The value of \(x\) is \(.....\)

A wire of resistance \(R\) and length \(L\) is cut into 5 equal parts. If these parts are joined parallely, then resultant resistance will be _______.

In the given circuit of potentiometer, the potential difference \(E\) across \(AB\) ( \(10\, m\) length) is larger than \(E _{1}\) and \(E _{2}\) as well. For key \(K _{1}\) (closed), the jockey is adjusted to touch the wire at point \(J_{1}\) so that there is no deflection in the galvanometer. Now the first battery \(\left( E _{1}\right)\) is replaced by second battery \(\left( E _{2}\right)\) for working by making \(K _{1}\) open and \(K _{2}\) closed. The galvanometer gives then null deflection at \(J _{2}\). The value of \(\frac{ E _{1}}{ E _{2}}\) is \(\frac{ a }{ b },\) where \(a =\) ...............

The electric potential at the centre of two concentric half rings of radii \(R_1\) and \(R_2\), having same linear charge density \(\lambda\) is

Two simple pendulums of length \(1\, m\) and \(4\, m\) respectively are both given small displacement in the same direction at the same instant. They will be again in phase after the shorter pendulum has completed number of oscillations equal to

Consider two solid spheres of radii \(\mathrm{R}_{1}=1 \;\mathrm{m}\) \(\mathrm{R}_{2}=2\; \mathrm{m}\) and masses \(\mathrm{M}_{1}\) and \(\mathrm{M}_{2},\) respectively. The gravitational field due to sphere \((1)\) and \((2)\) are shown. The value of \(\frac{\mathrm{M}_{1}}{\mathrm{M}_{2}}\) is