The amount of work done to break a big water drop of radius ' \(R\) ' into 27 small drops of equal radius is 10 J . The work done required to break the same big drop into 64 small drops of equal radius will be

A small bob tied at one end of a thin string of length \(1\, m\) is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio \(5: 1 .\) The velocity of the bob at the height position is ............. \(m / s\) (Take \(\left.g=10 \,m / s ^{2}\right)\)

When one light ray is reflected from a plane mirror with \(30^{\circ}\) angle of reflection, the angle of deviation of the ray after reflection is \(.....^{\circ}\)

In a parallel plate capacitor set up, the plate area of capacitor is \(2 \,m ^{2}\) and the plates are separated by \(1\, m\). If the space between the plates are filled with a dielectric material of thickness \(0.5\, m\) and area \(2\, m ^{2}\) (see \(fig.\)) the capacitance of the set-up will be \(.........\, \varepsilon_{0}\) (Dielectric constant of the material \(=3.2\) ) and (Round off to the Nearest Integer)

A cylinder with fixed capacity of \(67.2\, lit\) contains helium gas at \(STP\). The amount of heat needed to raise the temperature of the gas by \(20\,^oC\) is ..... \(J\) [Given that \(R = 8.31\, J\, mol^{-1}\, K^{-1}\)]

In a radioactive decay chain, the initial nucleus is \({}_{90}^{232}Th\).  At the end there are \(6\,\,\alpha -\) particles and \(4\,\,\beta -\) particles with are emitted. If the end nucleus is \({}_Z^AX\,,\,A\) and \(Z\) are given by

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be \(t_1\) and \(t_2\), respectively, then...

The distance between two plates of a capacitor is \(d\) and its capacitance is \(C _1\), when air is the medium between the plates. If a metal sheet of thickness \(\frac{2 d }{3}\) and of same area as plate is introduced between the plates, the capacitance of the capacitor becomes \(C _2\). The ratio \(\frac{ C _2}{ C _1}\) is:

A monochromatic light is incident on a metallic plate having work function \(\phi\). An electron, emitted normally to the plate from a point \(A\) with maximum kinetic energy, enters a constant magnetic field, perpendicular to the initial velocity of electron. The electron passes through a curve and hits back the plate at a point \(B\). The distance between A and B is :
(Given : The magnitude of charge of an electron is e and mass is \(\mathrm{m}, \mathrm{h}\) is Planck's constant and c is velocity of light. Take the magnetic field exists throughout the path of electron)

The trajectory of projectile, projected from the ground is given by \(y=x-\frac{x^2}{20}\). Where \(x\) and \(y\) are measured in meter. The maximum height attained by the projectile will be \(...........\,m\)

A particle performs simple harmonic motion with amplitude \(A\). Its speed is trebled at the instant that it is at a distance \(\frac{{2A}}{3}\) from equilibrium position. The new amplitude of the motion is

Glycerine of density \(1.25 \times 10^3\,kg\,m ^{-3}\) is flowing through the conical section of pipe. The area of cross-section of the pipe at its ends is \(10\,cm ^2\) and \(5\,cm ^2\) and pressure drop across its length is \(3\,Nm ^{-2}\). The rate of flow of glycerine through the pipe is \(x \times 10^{-5} m ^3 s ^{-1}\). The value of \(x\) is \(..............\).