A simple harmonic oscillator has an amplitude \(A\) and time period \(6 \pi\) second. Assuming the oscillation starts from its mean position, the time required by it to travel from \(x=A\) to \(x=\frac{\sqrt{3}}{2} A\) will be \(\frac{\pi}{x}\) s, where \(x=\) _______.

A circuit element \(X\) when connected to an a.c. supply of peak voltage \(100\,V\) gives a peak current of \(5\,A\) which is in phase with the voltage. A second element \(Y\) when connected to the same a.c. supply also gives the same value of peak current which lags behind the voltage by \(\frac{\pi}{2}\). If \(X\) and \(Y\) are connected in series to the same supply, what will be the rms value of the current in ampere?

A circular conducting coil of radius \(1\, {m}\) is being heated by the change of magnetic field \(\vec {B}\) passing perpendicular to the plane in which the coil is laid. The resistance of the coil is \(2\, \mu\, \Omega\). The magnetic field is slowly switched off such that its magnitude changes in time as \(B=\frac{4}{\pi} \times 10^{-3}\, {T}\left(1-\frac{{t}}{100}\right)\) The energy dissipated by the coil before the magnetic field is switched off completely is \(E=.....\,{m} J.\)

A proton of mass \(m\) collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of \(90^o\) with respect to each other. The mass of unknown particle is

Two short magnetic dipoles \({m}_{1}\) and \({m}_{2}\) each having magnetic moment of \(1\, {Am}^{2}\) are placed at point \({O}\) and \(P\) respectively. The distance between \(OP\) is \(1\, meter.\) The torque experienced by the magnetic dipole \({m}_{2}\) due to the presence of \({m}_{1}\) is ...... \(\times 10^{-7}\, {Nm}\)

The two wires \(A\) and \(B\) of equal cross-section but of different materials are joined together. The ratio of Young's modulus of wire \(A\) and wire \(B\) is \(20/11\). When the joined wire is kept under certain tension the elongations in the wires \(A\) and \(B\) are equal. If the length of wire \(A\) is \(2.2\) m, then the length of wire \(B\) is _______ m.

A circular disc has radius \(R_1\) and thickness \(T_1\). Another circular disc made of the same material has radius \(R_2\) and thickness \(T_2\). If the moment of inertia of both discs are same and \(\frac{R_1}{R_2}=2\) then \(\frac{T_1}{T_2}=\frac{1}{\alpha}\).The value of \(\alpha\) is __________ .

An unpolarized light beam is incident on the polarizer of a polarization experiment and the intensity of light beam emerging from the analyzer is measured as \(100\) Lumens. Now, if the analyzer is rotated around the horizontal axis (direction of light) by \(30^{\circ}\) in clockwise direction, the intensity of emerging light will be .......... Lumens.

A radioactive nucleus decays by two different processes. The half life for the first process is \(10\, s\) and that for the second is \(100 s\). the effective half life of the nucleus is close to\(.....sec\)

The mean intensity of radiation on the surface of the Sun is about \(10^{8}\,W/m^2.\) The \(rms\) value of the corresponding magnetic field is closet to