A radioactive substance decays to \(\left(\frac{1}{16}\right)^{t h}\) of its initial activity in \(80\, days.\) The half life of the radioactive substance expressed in days is ... .

A person of \(80\, kg\) mass is standing on the rim of a circular platform of mass \(200\, kg\) rotating about its axis as \(5\) revolutions per minute \((rpm).\) The person now starts moving towards the centre of the platform. What will be the rotational speed (in \(rpm\) ) of the platform when the person reaches its centre \(......rpm\)

In a metre-bridge when a resistance in the left gap is \(2\ \Omega\) and unknown resistance in the right gap, the balance length is found to be \(40\ \mathrm{cm}\). On shunting the unknown resistance with \(2\ \Omega\), the balance length changes by _______.

An engine is attached to a wagon through a shock absorber of length \(1.5\, {m}\). The system with a total mass of \(40,000\, {kg}\) is moving with a speed of \(72\, {kmh}^{-1}\) when the brakes are applied to bring it to rest. In the process of the system being brought to rest, the spring of the shock absorber gets compressed by \(1.0\, {m}\). If \(90\, \%\) of energy of the wagon is lost due to friction, the spring constant is \(....\, \times 10^{5}\, {N} / {m}\)

A monochromatic beam of light has a frequency \(v = \frac{3}{{2\pi }} \times {10^{12}}\,Hz\) and is propagating along the direction \(\frac{{\hat i + \hat j}}{{\sqrt 2 }}\). It is polarized along the \(\hat k\) direction. The acceptable form for the magnetic field is

A circular disc has radius \(R_1\) and thickness \(T_1\). Another circular disc made of the same material has radius \(R_2\) and thickness \(T_2\). If the moment of inertia of both discs are same and \(\frac{R_1}{R_2}=2\) then \(\frac{T_1}{T_2}=\frac{1}{\alpha}\).The value of \(\alpha\) is __________ .

The distance between charges \(+\mathrm{q}\) and \(-\mathrm{q}\) is \(2 l\) and between \(+2 \mathrm{q}\) and \(-2 \mathrm{q}\) is \(4 l\). The electrostatic potential at point \(P\) at a distance \(r\) from centre \(O\) is \(-\alpha\left[\frac{q l}{r^2}\right] \times 10^9 \mathrm{~V}\), where the value of \(\alpha\) is _______. (Use \(\left.\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\right)\)

An electron of a hydrogen like atom, having \(Z=4\), jumps from \(4^{\text {th }}\) energy state to \(2^{\text {nd }}\) energy state, The energy released in this process, will be \(.........eV\) \((\) Given \(Rch =13.6\,eV )\) Where \(R =\) Rydberg constant \(c=\) Speed of light in vacuum \(h =\) Planck's constant

If minimum possible work is done by a refrigerator in converting \(100\; grams\) of water at \(0^{\circ} C\) to ice, how much heat (in calories) is released to the surrounding at temperature \(27^{\circ} C\) (Latent heat of ice \(=80 Cal / gram\) ) to the nearest integer?

Seven identical circular planar disks, each of mass \(M\) and radius \(R\) are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point \(P\) is:

A piston of mass \(M\) is hung from a massless spring whose restoring force law goes as \(\mathrm{F}=-\mathrm{kx}^3\), where k is the spring constant of appropriate dimension. The piston separates the vertical chamber into two parts, where the bottom part is filled with ' \(n\) ' moles of an ideal gas. An external work is done on the gas isothermally (at a constant temperature T) with the help of a heating filament (with negligible volume) mounted in lower part of the chamber, so that the piston goes up from a height \(\mathrm{L}_0\) to \(\mathrm{L}_1\), the total energy delivered by the filament is (Assume spring to be in its natural length before heating)