Two liquids of densities \(\rho_{1}\) and \(\rho_{2}\left(\rho_{2}=2 \rho_{1}\right)\) are filled up behind a square wall of side \(10\; \mathrm{m}\) as shown in figure. Each liquid has a height of \(5 \;\mathrm{m} .\) The ratio of the forces due to these liquids exerted on upper part \(MN\) to that at the lower pait \(NO\) is (Assume that the liquids are not mixing)

A child of mass \(5\,kg\) is going round a merry-goround that makes \(1\) rotation in \(3.14\,s\). The radius of the merry-go-round is \(2\,m\). The centrifugal force on the child will be \(.......\,N\)

The electric fields of two plane electromagnetic plane waves in vacuum are given by \(\overrightarrow{\mathrm{E}}_{1}=\mathrm{E}_{0} \hat{\mathrm{j}} \cos (\omega \mathrm{t}-\mathrm{kx})\) and \(\overrightarrow{\mathrm{E}}_{2}=\mathrm{E}_{0} \hat{\mathrm{k}} \cos (\omega \mathrm{t}-\mathrm{ky})\) At \(t=0,\) a particle of charge \(q\) is at origin with a velocity \(\overrightarrow{\mathrm{v}}=0.8 \mathrm{c} \hat{\mathrm{j}}\) (\(c\) is the speed of light in vacuum). The instantaneous force experienced by the particle is 

A parallel plate capacitor is formed by two plates each of area \(30 \pi\, cm ^{2}\) separated by \(1\, mm\). A material of dielectric strength \(3.6 \times 10^{7} \,Vm ^{-1}\) is filled between the plates. If the maximum charge that can be stored on the capacitor without causing any dielectric breakdown is \(7 \times 10^{-6}\, C\), the value of dielectric constant of the material is \(\left\{ Use : \frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} Nm ^{2} C ^{-2}\right\}\)

A spherical body of mass \(100 \mathrm{~g}\) is dropped from a height of \(10 \mathrm{~m}\) from the ground. After hitting the ground, the body rebounds to a height of \(5 \mathrm{~m}\). The impulse of force imparted by the ground to the body is given by : _______. (given \(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2\) )

A cylindrical conductor of length 2m and area of cross-section \( 0.2\,mm^{2} \) carries an electric current of 1.6 A when its ends are connected to a 2V battery. Mobility of electrons in the conductor is \( \alpha\times10^{-3} m^{2}/V.s \). The value of \( \alpha \) is:
(electron concentration \( =5\times10^{28}/m^{3} \) and electron charge \( =1.6\times10^{-19} C \))

In an experiment to measure focal length \((f)\) of convex lens, the least counts of the measuring scales for the position of object \((u)\) and for the position of image \((v)\) are \(\Delta u\) and \(\Delta v\), respectively. The error in the measurement of the focal length of the convex lens will be _______.

A paralle plate capacitor is made up of stair like structure with a palte area \(A\) of each stair and that is connected with a wire of length \(b\), as shown in the figure. The capacitance of the arrangement is \(\frac{ x }{15} \frac{\varepsilon_{0} A }{ b }\). The value of \(x\) is ............

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of \(\theta ,\) where \(\theta \) is the angle by which it has rotated, is given as \(k\theta ^2.\) If its moment of inertia is \(I\) then the angular acceleration of the disc is

A small block slides down from the top of hemisphere of radius \(R=3\, {m}\) as shown in the figure. The height \('h'\) at which the block will lose contact with the surface of the sphere is \(.............  \;\;m\). (Assume there is no friction between the block and the hemisphere)