A mercury drop of radius \(10^{-3}\,m\) is broken into \(125\) equal size droplets. Surface tension of mercury is \(0.45\,Nm ^{-1}\). The gain in surface energy is \(......\times 10^{-5}\,J\)

A certain charge \(Q\) is divided into two parts \(q\) and \((Q-q) .\) How should the charges \(Q\) and \(q\) be divided so that \(q\) and \((Q-q)\) placed at a certain distance apart experience maximum electrostatic repulsion?

The time period of simple harmonic motion of mass \(\mathrm{M}\) in the given figure is \(\pi \sqrt{\frac{\alpha M}{5 K}}\), where the value of \(\alpha\) is _______.

Nearly \(10 \%\) of the power of a \(110\,W\) light bulb is converted to visible radiation. The change in average intensities of visible radiation, at a distance of \(1\, m\) from the bulb to a distance of \(5\,m\) is \(a \times 10^{-2}\,W / m ^{2}\). The value of ' \(a\) ' will be.

In the following circuit, the magnitude of current \(I _{ l }\), is \(............A\).

A \(70\, kg\) man leaps vertically into the air from a crouching position. To take the leap the man pushes the ground with a constant force \(F\) to raise himself The center of gravity rises by \(0.5\, m\) before he leaps. After the leap the \(c.g.\) rises by another \(1\, m\). The maximum power delivered by the muscles is : (Take \(g\, = 10\, ms^{-2}\))

An insulated copper wire of \(100\) turns is wrapped around a wooden cylindrical core of the crosssectional area \(24\,cm ^2\). The two ends of the wire are connected to a resistor. The total resistance in the circuit is \(12\,\Omega\). If an externally applied uniform magnetic field in the core along its axis changes from \(1.5 T\) in one direction to \(1.5\,T\) in the opposite direction, the charge flowing through a point in the circuit during the change of magnetic field will be \(.........\,mC\).

At a given point of time the value of displacement of a simple harmonic oscillator is given as \(y = A \cos \left(30^{\circ}\right)\). If amplitude is \(40\,cm\) and kinetic energy at that time is \(200\, J\), the value of force constant is \(1.0 \times 10^{ x }\,Nm ^{-1}\). The value of \(x\) is ......

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.

Both the nucleus and the atom of some element are in their respective first excited states. They get de-excited by emitting photons of wavelengths \(\lambda _N,\,\lambda _A\)  respectively. The ratio \(\frac{{{\lambda _N}}}{{{\lambda _A}}}\)  is closest to

The percentage decrease in the weight of a rocket, when taken to a height of \(32 km\) above the surface of earth will, be\(.....\%\) (Radius of earth \(=6400\,km\) )