A stone of mass \(1\,kg\) is tied to end of a massless string of length \(1\,m\). If the breaking tension of the string is \(400\,N\), then maximum linear velocity, the stone can have without breaking the string, while rotating in horizontal plane, is \(.......\,ms^{-1}\)

If momentum \([ P ]\), area \([ A ]\) and time \([ T ]\) are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :

An ideal monoatomic gas is confined in a cylinder by a spring loaded piston of cross section \(8.0\times10^{-3}\, m^2\) . Initially the gas is at \(300\, K\) and occupies a volume of \(2.4\times10^{-3}\, m^3\) and the spring is in its relaxed state as shown in figure. The gas is heated by a small heater until the piston moves out slowly by \(0.1\, m\). The force constant of the spring is \(8000\, N/m\) and the atmospheric pressure is \(1.0\times10^5\, N/m^2\) . The cylinder and the piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and the cylinder. The final temperature of the gas will be: (Neglect the heat loss through the lead wires of the heater . The heat capacity of the heater coil is also negligible)

A series combination of resistor of resistance \(100\,\Omega\), inductor of inductance \(1\,H\) and capacitor of capacitance \(6.25\,\mu F\) is connected to an ac source. The quality factor of the circuit will be \(.............\).

Two capacitors of capacities \(2 {C}\) and \({C}\) are joined in parallel and charged up to potential \({V}\). The battery is removed and the capacitor of capacity \(C\) is filled completely with a medium of dielectric constant \({K}\). The potential difference across the capacitors will now be

Two blocks (\(P\) and \(Q\)) with respectively masses \(2\) kg and \(1.5\) kg are joined by a massless thread. These blocks are mounted on a frictionless pulley which is fixed on the edge of a cube (\(S\)), as shown in the figure below. Block \(P\) is positioned on the top surface which has no friction and block \(Q\) is in contact with side-surface, having coefficient friction \(\mu\). The cube (\(S\)) moves towards the right with acceleration of \(\dfrac{g}{2}\), where \(g\) is gravitational acceleration. During this movement the block \(P\) and \(Q\) remain stationary. The value of \(\mu\) is _______. (take \(g = 10\) m/s\(^2\))

The parallel combination of two air filled parallel plate capacitors of capacitance \(C\) and \(nC\) is connected to a battery of voltage, \(V\). When the capacitor are fully charged, the battery is removed and after that a dielectric material of dielectric constant \(K\) is placed between the two plates of the first capacitor. The new potential difference of the combined system is

With rise in temperature, the Young's modulus of elasticity _______.

A simple pendulum doing small oscillations at a place \(\mathrm{R}\) height above earth surface has time period of \(T_1=4 \mathrm{~s}\). \(T_2\) would be it's time period if it is brought to a point which is at a height \(2 R\) from earth surface. Choose the correct relation. \([R=\) radius of Earth]:

Two charges \(7 \mu \mathrm{c}\) and \(-4 \mu \mathrm{c}\) are placed at \((-7 \mathrm{~cm}, 0,0)\) and \((7 \mathrm{~cm}, 0,0)\) respectively. Given, \(\epsilon_0=8.85 \times 10^{-12} \mathrm{C}^2 \mathrm{~N}^{-1} \mathrm{~m}^{-2}\), the electrostatic potential energy of the charge configuration is :

If \(R , X _{ L }\). and \(X _{ C }\) represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless:

When \(100\,g\) of a liquid \(A\) at \(100\,^oC\) is added to \(50\,g\) of a liquid \(B\) at temperature \(75\,^oC\), the temperature of the mixture becomes \(90\,^oC\). The temperature of the mixture, if \(100\,g\) of liquid \(A\) at \(100\,^oC\) is added to \(50\,g\) of liquid \(B\) at \(50\,^oC\), will be ........\(^oC\)

A simple pendulum is being used to determine the value of gravitational acceleration \(\mathrm{g}\) at a certain place. The length of the pendulum is \(25.0\; \mathrm{cm}\) and a stop watch with \(1\; \mathrm{s}\) resolution measures the time taken for \(40\) oscillations to be \(50\; s\). The accuracy in \(g\) is  ....... \(\%\)