A sample of gas at temperature \(\mathrm{T}\) is adiabatically expanded to double its volume. Adiabatic constant for the gas is \(\gamma=3 / 2\). The work done by the gas in the process is _______. \((\mu=1 \mathrm{~mole})\)

The threshold frequency of metal is \(f_0\). When the light of frequency \(2 f_0\) is incident on the metal plate, the maximum velocity of photoelectron is \(v_1\). When the frequency of incident radiation is increased to \(5 f _0\). the maximum velocity of photoelectrons emitted is \(v_2\). The ratio of \(v_1\) to \(v_2\) is:

The time dependence of the position of a particle of mass \(m = 2\) is given by \(\vec r\,(t)\, = \,2t\,\hat i\, - 3{t^2}\hat j\) Its angular momentum with respect to the origin at time \(t = 2\) is

In the circuit shown, find \(C\) if the effective capacitance of the whole circuit is to be \(0.5\,\mu F.\) All values in the circuit are in \(\mu F.\)

Two identical charged particles each having a mass \(10 \,g\) and charge \(2.0 \times 10^{-7}\,C\) area placed on a horizontal table with a separation of \(L\) between then such that they stay in limited equilibrium. If the coefficient of friction between each particle and the table is \(0.25\), find the value of \(L\).[Use \(g =10\,ms ^{-2}\) ]..........\(cm\)

When a polaroid sheet is rotated between two crossed polaroids then the transmitted light intensity will be maximum for a rotation of :

As shown in the figure, a block of mass \(\sqrt{3}\, kg\) is kept on a horizontal rough surface of coefficient of friction \(\frac{1}{3 \sqrt{3}}\). The critical force to be applied on the vertical surface as shown at an angle \(60^{\circ}\) with horizontal such that it does not move, will be \(3 x\). The value of \(3x\) will be \(\left[ g =10 m / s ^{2} ; \sin 60^{\circ}=\frac{\sqrt{3}}{2} ; \cos 60^{\circ}=\frac{1}{2}\right]\)

A plane electromagnetic wave with frequency of \(30 {MHz}\) travels in free space. At particular point in space and time, electric field is \(6 {V} / {m}\). The magnetic field at this point will be \({x} \times 10^{-8} {T}\). The value of \({x}\) is ..... .

A vessel contains \(16 \,g\) of hydrogen and \(128 \,g\) of oxygen at standard temperature and pressure. The volume of the vessel in \(cm ^{3}\) is

If \(\mathrm{n}\) is the number density and \(\mathrm{d}\) is the diameter of the molecule, then the average distance covered by a molecule between two successive collisions (i.e. mean free path) is represented by _______.

Voltage rating of a parallel plate capacitor is \(500\,V\). Its dielectric can withstand a maximum electric field of \({10^6}\,\frac{V}{m}\). The plate area is \(10^{-4}\, m^2\) . What is the dielectric constant if the capacitance is \(15\, pF\) ? (given \({ \in _0} = 8.86 \times {10^{ - 12}}\,{C^2}\,/N{m^2}\))