A spherical shell of \(1 \,kg\) mass and radius \(R\) is rolling with angular speed \(\omega\) on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin \(O\) is \(\frac{a}{3} R^{2} \omega\). The value of a will be ..............

The focal length of the objective and the eyepiece of a telescope are \(50\,cm\) and \(5\,cm\) respectively. If the telescope is focussed for distinct vision on a scale distant \(2\,m\) from its objective, then its magnifying power will be

On celcius scale the temperature of body increases by \(40^{\circ} \mathrm{C}\). The increase in temperature on Fahrenheit scale is _______.

An alternating voltage source \(V =260 \sin (628\,t )\) is connected across a pure inductor of \(5\,mH\). Inductive reactance in the circuit is \(......... \Omega\)

Choose the correct nuclear process from the below options.
[p: proton, n : neutron, \(\mathrm{e}^{-}\): electron, \(\mathrm{e}^{+}\): positron, \(v:\) neutrino, \(\bar{v}\) : antineutrino]

A gas molecule of mass \(M\) at the surface of the Earth has kinetic energy equivalent to \(0\,^oC\). If it were to go up straight without colliding with any other molecules, how high it would rise? Assume that the height attained is much less than radius of the earth. (\(k_B\) is Boltzmann constant)

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet.
Reason (R): The mass of the pendulum remains unchanged at Earth and the other planet.
In the light of the above statements, choose the correct answer from the options given below :

A ray of light entering from air into a denser medium of refractive index \(\frac{4}{3}\), as shown in figure. The light ray suffers total internal reflection at the adjacent surface as shown. The maximum, value of angle \(\theta\) should be equal to -

A potential divider circuit is shown in figure. The output voltage \(V_0\) is _______.

The energy equivalent of \(1 \mathrm{~g}\) of substance is _______.

Two waves of intensity ratio \(1: 9\) cross each other at a point. The resultant intensities at the point, when (a) Waves are incoherent is \(I_1\) (b) Waves are coherent is \(I_2\) and differ in phase by \(60^{\circ}\). If \(\frac{I_1}{I_2}=\frac{10}{x}\), then \(x=\) _______.

The box of a pin hole camera, of length \(L\), has a hole of radius \(a\). It is assumed that when the hole is illuminated by a parallel beam of light of wavelength \(\lambda\), the spread of the spot (obtained on the opposite wall of the camera) is the sum of its geometrical spread and the spread due to diffraction. The spot would then have its minimum size ( say \(b_{min}\)) when

In the given figure, the charge stored in \(6 \ \mu \mathrm{F}\) capacitor, when points \(A\) and \(B\) are joined by a connecting wire is _______ \(\mu \mathrm{C}\).