Time period of a simple pendulum is \(T\). The time taken to complete \(5 / 8\) oscillations starting from mean position is \(\frac{\alpha}{\beta} T\). The value of \(\alpha\) is ..... .

A charge of \(4.0 \mu \mathrm{C}\) is moving with a velocity of \(4.0 \times 10^6 \mathrm{~ms}^{-1}\) along the positive \(\mathrm{y}\)-axis under a magnetic field \(\vec{B}\) of strength \((2 \hat{k})\  T\). The force acting on the charge is \(x \hat{i} N\). The value of \(x\) is _______.

Two slabs with square cross section of different materials \((1,2)\) with equal sides \((l)\) and thickness \(\mathrm{d}_1\) and \(\mathrm{d}_2\) such that \(\mathrm{d}_2=2 \mathrm{~d}_1\) and \(l \gt \mathrm{d}_2\). Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of deformation is \(\theta_2=2 \theta_1\). If the shear moduli of material 1 is \(4 \times 10^9 \mathrm{~N} / \mathrm{m}^2\), then shear moduli of material 2 is \(\mathrm{x} \times 10^9 \mathrm{~N} / \mathrm{m}^2\), where value of \(x\) is ________.

Diameter of a plano-convex lens is \(6 \,cm\) and thickness at the centre is \(3\, mm\). If the speed of light in the material of the lens is \(2  \times  10^8 m/sec\), the focal length of the lens is.......\(cm\)

Given below are two statements: Statement \(I\) : The diamagnetic property depends on temperature. Statement \(II\) : The included magnetic dipole moment in a diamagnetic sample is always opposite to the magnetizing field. In the light of given statement, choose the correct answer from the options given below:

A uniform sphere of mass \(500\; g\) rolls without slipping on a plane horizontal surface with its centre moving at a speed of \(5.00\; \mathrm{cm} / \mathrm{s}\). Its kinetic energy is

A tuning fork of frequency \(340\,Hz\) resonates in the fundamental mode with an air column of length \(125\,cm\) in a cylindrical tube closed at one end. When water is slowly poured in it, the minimum height of water required for observing resonance once again is________ \(cm\) (Velocity of sound in air is \(340\,ms ^{-1}\) )

An electron moving with speed \(v\) and a photon moving with speed \(c,\) have same D-Broglie wavelength. The ratio of kinetic energy of electron to that of photon is -

For a hydrogen atom, the ratio of the largest wavelength of Lyman series to that of the Balmer series is _______.