Moment of inertia of a cylinder of mass \(M,\) length \(L\) and radius \(R\) about an axis passing through its centre and perpendicular to the axis of the cylinder is \(I = M \left(\frac{ R ^{2}}{4}+\frac{ L ^{2}}{12}\right) .\) If such a cylinder is to be made for a given mass of material, the ratio \(\frac LR\) for it to have minimum possible \(I\) is

What equal length of an iron wire and a copper-nickel alloy wire, each of \(2 \; {mm}\) diameter connected parallel to give an equivalent resistance of \(3 \Omega ?\) (Given resistivities of iron and copper-nickel alloy wire are \(12 \;\mu \Omega {cm}\) and \(51\; \mu \Omega {cm}\) respectively) (in \({m}\))

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)

If \(\epsilon_0\) denotes the permittivity of free space and \(\Phi_{\mathrm{E}}\) is the flux of the electric field through the area bounded by the closed surface, then dimension of \(\left(\epsilon_0 \frac{\mathrm{~d} \phi_{\mathrm{E}}}{\mathrm{dt}}\right)\) are that of :

An unpolarised light beam of intensity \(2 I _{0}\) is passed through a polaroid \(P\) and then through another polaroid \(Q\) which is oriented in such a way that its passing axis makes an angle of \(30^{\circ}\) relative to that of \(P\). The intensity of the emergent light is.

A flexible chain of mass \(m\) hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is \(30^{\circ}\). Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is _________ .

In Young's double slit experiment the two slits are \(0.6 \; mm\) distance apart. Interference pattern is observed on a screen at a distance \(80 \; cm\) from the slits. The first dark fringe is observed on the screen directly opposite to one of the slits. The wavelength of light will be \(\dots \; nm .\)

An electron and proton are separated by a large distance. The electron starts approaching the proton with energy \(3\, {eV}\). The proton captures the electrons and forms a hydrogen atom in second excited state. The resulting photon is incident on a photosensitive metal of threshold wavelength \(4000\, \mathring {{A}}\). What is the maximum kinetic energy of the emitted photoelectron ? (In \({eV}\))

An alternating voltage of amplitude \(40 \mathrm{~V}\) and frequency \(4 \mathrm{kHz}\) is applied directly across the capacitor of \(12 \mu \mathrm{F}\). The maximum displacement current between the plates of the capacitor is nearly _______.

Two radioactive materials \(A\) and \(B\) have decay constants \(10\,\lambda \) and \(\lambda \), respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of a to that of \(B\) will be \(1/e\) after a time

Three rods of Copper, Brass and Steel are welded together to form a \(Y\) shaped structure. Area of cross - section of each rod \(= 4\ cm^2\) . End of copper rod is maintained at \(100^o C \) where as ends ofbrass and steel are kept at \(0^o C\). Lengths of the copper, brass and steel rods are \(46, 13\) and \(12\ cms\) respectively. The rods are thermally insulated from surroundings excepts at ends. Thermal conductivities of copper, brass and steel are \(0.92, 0.26\) and \(0.12\ CGS\) units respectively. Rate ofheat flow through copper rod is ....... \(cal\, s^{-1}\)

The dimension of stopping potential \(\mathrm{V}_{0}\) in photoelectric effect in units of Planck's constant \(h\), speed of light \(c\) and Gravitational constant \(G\) and ampere \(A\) is