A ring and a solid sphere roll down the same inclined plane without slipping. They start from rest. The radii of both bodies are identical and the ratio of their kinetic energies is \(\frac{7}{\mathrm{x}}\) where \(\mathrm{x}\) is___________.

The frequency of revolution of the electron in Bohr's orbit varies with \(n\), the principal quantum number as _______.

A parallel plate capacitor has plates of area \(A\) separated by distance \(d\) between them. It is filled with a dielectric which has a dielectric constant that varies as \(\mathrm{k}(\mathrm{x})=\mathrm{K}(1+\alpha \mathrm{x})\) where \(\mathrm{x}\) is the distance measured from one of the plates. If \((\alpha \text {d)}<<1,\) the total capacitance of the system is best given by the expression 

Water fall from a \(40\,m\) high dam at the rate of \(9 \times 10^{4}\,kg\) per \(hour\). Fifty percentage of gravitational potential energy can be converted into electrical energy. Using this hydroelectric energy number of \(100\,W\) lamps, that can be lit, is (Take \(\left.g=10\,ms ^{-2}\right)\)

A clock has a continuously moving second's hand of \(0.1\, m\) length. The average acceleration of the tip of the hand (in units of \(ms ^{-2}\) ) is of the order of

A positive ion \(A\) and a negative ion \(B\) has charges \(6.67 \times 10^{-19} \mathrm{C}\) and \(9.6 \times 10^{-10} \mathrm{C}\), and masses \(19.2 \times 10^{-27} \mathrm{~kg}\) and \(9 \times 10^{-27} \mathrm{~kg}\) respectively. At an instant, the ions are separated by a certain distance r. At that instant the ratio of the magnitudes of electrostatic force to gravitational force is \(P \times 10^{45}\), where the value of 10\(P\) is _______.
(Take \(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-1}\) and universal gravitational constant as \(6.67 \times 10^{-11} \mathrm{Nm}^2 \mathrm{~kg}^{-2}\) )
Assume that charge may not be an integral multiple of electrons.

In interference experiment the path difference between two interfering waves at a point \(A\) on the screen is \(\lambda/3\), where \(\lambda\) is the wavelength of these waves, and at another point \(B\) the path difference is \(\lambda/6\). The ratio of intensities at points \(A\) and \(B\) is _______.

Heat is supplied to a diatomic gas at constant pressure. Then the ratio of \(\Delta Q : \Delta U : \Delta W\) is _______.

An object viewed from a near point distance of \(25\, {cm}\), using a microscopic lens with magnification \('6',\) gives an unresolved image. A resolved image is observed at infinite distance with a total magnification double the earlier using an eyepiece along with the given lens and a tube of length \(0.6\, {m}\), if the focal length of the eyepiece is euqal to \(.... \,{cm} .\)

In an experiment, electrons are accelerated, from rest, by applying, a voltage of \(500 \,V.\) Calculate the radius of the path if a magnetic field \(100\,mT\) is then applied. [Charge of the electron \(= 1.6 \times 10^{-19}\,C\) Mass of the electron \(= 9.1 \times 10^{-31}\,kg\) ]

The acceleration of an electron in the first orbit of the hydrogen atom \((z\, = 1)\) is