The angle between vector \(\vec{Q}\) and the resultant of \((2 \overrightarrow{\mathrm{Q}}+2 \overrightarrow{\mathrm{P}})\) and \((2 \overrightarrow{\mathrm{Q}}-2 \overrightarrow{\mathrm{P}})\) is _______.

The time period of revolution of electron in its ground state orbit in a hydrogen atom is \(1.6 \times 10^{-16} \;\mathrm{s}\). The frequency of revolution of the electron in its first excited state (in \(s^{-1}\) ) is:

A body starts moving from rest with constant acceleration covers displacement \(S_1\) in first \((p-1)\) seconds and \(S_2\) in first \(p\) seconds. The displacement \(S_1+S_2\) will be made in time :

A black coloured solid sphere of radius \(R\) and mass \(M\) is inside a cavity with vacuum inside. The walls of the cavity are maintained at temperature \(T_0\). The initial temperature of the sphere is \(3T_0\). If the specific heat of the material of the sphere varies as \(\alpha T^3\) per unit mass with the temperature \(T\) of the sphere, where \(\alpha \) is a constant, then the time taken for the sphere to cool down to temperature \(2T_0\) will be ( \(\sigma \) is Stefan Boltzmann constant)

A small ball of mass \(m\) and density \(\rho\) is dropped in a viscous liquid of density \(\rho_0\). After sometime, the ball falls with constant velocity. The viscous force on the ball is _______.

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  ....... \(\%\)

A \(5\) mg particle carrying a charge of \(5\pi\times 10^{-6}\) C is moving with velocity of \((3\hat{i}+2\hat{k})\times 10^{-2}\) m/s in a region having magnetic field \(\vec{B} = 0.1\hat{k}\) Wb/m\(^2\). It moves a distance of \(\alpha\) meter along \(\hat{k}\) when it completes \(5\) revolutions. The value of \(\alpha\) is ________.

On heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius \(R\) and making a circular contact of radius \(r\) with the bottom of the vessel. If \(r < < R\), and the surface tension of water is \(T\), value of \(r\) just before bubbles detach is (density of water is \(\rho_{w}\))

The activity of a radioactive sample falls from \(700 \;\mathrm{s}^{-1}\) to \(500\; \mathrm{s}^{-1}\) in \(30\;min\). Its half life is close to.........\(min\)

A small metallic sphere of diameter 2 mm and density \(10.5 g / cm ^3\) is dropped in glycerine having viscosity 10 Poise and density \(1.5 g / cm ^3\) respectively. The terminal velocity attained by the sphere is ___________ \(cm / s\).
\(\left(\pi=\frac{22}{7}\right.\) and \(\left.g=10 m / s ^2\right)\)