A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is \(\sqrt{\frac{2}{x}}\). The value of \(x\) is \(.......\)

An electric appliance supplies \(6000\, {J} / {min}\) heat to the system. If the system delivers a power of \(90\, {W}\). How long (in \(sec\)) it would take to increase the internal energy by \(2.5 \times 10^{3}\, {J}\) ?

Two cells are connected between points \(A\) and \(B\) as shown. Cell \(1\) has emf of \(12\,V\) and internal resistance of \(3\,\Omega\). Cell \(2\) has emf of \(6\,V\) and internal resistance of \(6\,\Omega\). An external resistor \(R\) of \(4\,\Omega\) is connected across \(A\) and \(B\). The current flowing through \(R\) will be \(.............A\).

When radiation of wavelength \(\lambda \) is incident on a metallic surface, the stopping potential is \(4.8\) volts. If the same surface is illuminated with radiation of double the wavelength, then the stopping potential becomes \(1.6\) volts. Then the threshold wavelength for the surface is

A diatomic gas \((\gamma=1.4)\) does \(400 J\) of work when it is expanded isobarically. The heat given to the gas in the process is ............ \(J\)

A wooden block floating in a bucket of water has \(\frac{4}{5}\) of its volume submerged. When certain amount of an oil is poured into the bucket, it is found that the block is just under the oil surface with half of its volume under water and half in oil. The density of oil relative to that of water is

The electric field a plane electromagnetic wave is given by :
\(E _y=69 \sin \left[0.6 \times 10^3 x -1.8 \times 10^{11} t \right] V / m\).
The expression for magnetic field associated with this electromagnetic wave is _________ T.

A plane electromagnetic wave travels in a medium of relative permeability \(1.61\) and relative permittivity \(6.44\). If magnitude of magnetic intensity is \(4.5 \times 10^{-2} \;Am ^{-1}\) at a point, what will be the approximate magnitude of electric field intensity at that point\(?\) (Given : permeability of free space \(\mu_{0}=4 \pi \times 10^{-7}\;NA ^{-2}\), speed of light in vacuum \(c =3 \times 10^{8} \;ms ^{-1}\) )

A copper wire of length \(1.0\, m\) and a steel wire of length \(0.5\, m\) having equal cross-sectional areas are joined end to end. The composite wire is stretched by a certain load which stretches the copper wire by \(1\, mm\). If the Young's modulii of copper and steel are respectively \(1.0\times10^{11}\, Nm^{-2}\) and \(2.0\times10^{11}\, Nm^{- 2}\), the total extension of the composite wire is ........ \(mm\)

The electrostatic potential in a charged spherical region of radius \(r\) varies as \(V=a r^3+b\), where \(a\) and b are constants. The total charge in the sphere of unit radius is \(\alpha \times \pi a \in_0\). The value of \(\alpha\) is _________ .

The length of wire becomes \(l_1\) and \(l_2\) when \(100\,N\) and \(120\,N\) tensions are applied respectively. If \(10l_2=11l_1\), the natural length of wire will be \(\frac{1}{x} l_1\). Here the value of \(x\) is ........