Two charged conducting spheres \(S_1\) and \(S_2\) of radii \(8\) cm and \(18\) cm are connected to each other by a wire. After equilibrium is established, the ratio of electric fields on \(S_1\) and \(S_2\) spheres are \(E_{S_1}\) and \(E_{S_2}\) respectively. The value of \(\dfrac{E_{S_1}}{E_{S_2}}\) is _______.

As shown in the figure a block of mass \(10\,kg\) lying on a horizontal surface is pulled by a force \(F\) acting at an angle \(30^{\circ}\), with horizontal. For \(\mu_{ s }=0.25\), the block will just start to move for the value of \(F..........\,N\) : \(\left[\right.\) Given \(\left.g =10\,ms ^{-2}\right]\)

Two equal positive point charges are separated by a distance \(2 a\). The distance of a point from the centre of the line joining two charges on the equatorial line (perpendicular bisector) at which force experienced by a test charge \(q_0\) becomes maximum is \(\frac{a}{\sqrt{x}}\). The value of \(x\) is \(................\)

Match List\(-I\) with List\(-II\).

List\(-I\)	List\(-II\)
\((A)\) Angular momentum	\((I)\) \(\left[ ML ^2 T ^{-2}\right]\)
\((B)\) Torque	\((II)\) \(\left[ ML ^{-2} T ^{-2}\right]\)
\((C)\) Stress	\((III)\) \(\left[ ML ^2 T ^{-1}\right]\)
\((D)\) Pressure gradient	\((IV)\) \(\left[ ML ^{-1} T ^{-2}\right]\)

Choose the correct answer from the options given below:

Two \(4\) bits binary numbers, \(A = 1101\) and \(B = 1010\) are given in the inputs of a logic circuit shown in figure below. The output \((Y)\) will be:

The motion of a simple pendulum excuting \(S.H.M\). is represented by following equation. \(Y = A \sin (\pi t +\phi)\), where time is measured in \(second\). The length of pendulum is .............\(cm\)

The angular momentum for the electron in Bohr's orbit is \(L\). If the electron is assumed to revolve in second orbit of hydrogen atom, then the change in angular momentum will be

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)

In a system two particles of masses \(m_1=3 \mathrm{~kg}\) and \(\mathrm{m}_2=2 \mathrm{~kg}\) are placed at certain distance from each other. The particle of mass \(m_1\) is moved towards the center of mass of the system through a distance \(2 \mathrm{~cm}\). In order to keep the center of mass of the system at the original position, the particle of mass \(\mathrm{m}_2\) should move towards the center of mass by the distance _______\(\mathrm{cm}\).

Hydrogen atom from excited state comes to the ground by emitting a photon of wavelength \(\lambda\). The value of principal quantum number ' \(n\) ' of the excited state will be: (\(R\): Rydberg constant)

Two masses 400 g and 350 g are suspended from the ends of a light string passing over a heavy pulley of radius 2 cm . When released from rest the heavier mass is observed to fall 81 cm in 9 s . The rotational inertia of the pulley is _________ \(kg \cdot m ^2\).\(\left( g =9.8 m / s ^2\right)\)

\(\left(P+\frac{a}{V^2}\right)(V-b)=R T\) represents the equation of state of some gases. Where \(P\) is the pressure, \(V\) is the volume, \(T\) is the temperature and \(a, b, R\) are the constants. The physical quantity, which has dimensional formula as that of \(\frac{b^2}{a}\), will be

In an experiment to verify Stokes law, a small spherical ball of radius \(r\) and density \(\rho\) falls under gravity through a distance \(h\) in air before entering a tank of water. If the terminal velocity of the ball inside water is same as its velocity just before entering the water surface, then the value of \(h\) is proportional to : (ignore viscosity of air)