A heat engine operates between a cold reservoir at temperature \({T}_{2}=400\, {K}\) and a hot reservoir at temperature \({T}_{1} .\) It takes \(300 \,{J}\) of heat from the hot reservoir and delivers \(240\, {J}\) of heat to the cold reservoir in a cycle. The minimum temperature of the hot reservoir has to be \(....{K}\)

A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should by unbinding the strings to achieve a speed of \(4\,ms ^{-1}\), is\(........cm\). \(\left(\right.\) take \(\left.g=10\,ms ^{-2}\right)\)

If \({R}_{{E}}\) be the radius of Earth, then the ratio between the acceleration due to gravity at a depth \(' {r} '\) below and a height \(' r '\) above the earth surface is: (Given : \(\left.{r}<{R}_{{E}}\right)\)

The ratio of speed of sound in hydrogen gas to the speed of sound in oxygen gas at the same temperature is

A water drop of radius \(1\,cm\) is broken into \(729\) equal droplets. If surface tension of water is \(75\,dyne / cm\), then the gain in surface energy upto first decimal place will be\(...\times 10^{-4}\,J\)

This question has Statement \(-1\) and Statement \(-2\) Of the four choices given after the Statements, choose the one that best describes the two Statements Statement \(1\) : No work is required to be done to move a test charge between any two points on an equipotential surface Statement \(2\) : Electric lines of force at the equipotential surfaces are mutually perpendicular to each other

Two particles \(\mathrm{X}\) and \(\mathrm{Y}\) having equal charges are being accelerated through the same potential difference. Thereafter they enter normally in a region of uniform magnetic field and describes circular paths of radius \(R_1\) and \(R_2\) respectively. The mass ratio of \(\mathrm{X}\) and \(\mathrm{Y}\) is _______.

Consider a cylinder of mass \(M\) resting on a rough horizontal rug that is pulled out from under it with acceleration \('a'\) perpendicular to the axis of the cylinder. What is \(F_{friction}\) at point \(P\) ? It is assumed that the cylinder does not slip

A particle of mass \({m}\) is suspended from a ceiling through a string of length \(L\). The particle moves in a horizontal circle of radius \(r\) such that \({r}=\frac{{L}}{\sqrt{2}}\). The speed of particle will be:

The energy of \(He ^{+}\)ion in its first excited state is \(......eV\) (The ground state energy for the Hydrogen atom is \(-13.6\,eV )\)

In the circuit shown, current (in \(A\)) through \(50\; V\) and \(30\; V\) batteries are, respectively

Muon \((\mu ^{-1})\) is negatively charged \(\left( {\left| q \right| = \left| e \right|} \right)\) with a mass \(m_{\mu}\, = 200\,m_e\), where \(m_e\) is the mass of the electron and \(e\) is the electronic charge. If \(\mu ^{-1}\) is bound to a proton to form a hydrogen like atom, identify the correct statements \((A)\) Radius of the muonic orbit is \(200\) times smaller than that of the electron \((B)\) the speed of the \(\mu ^{-1}\) in the \(n^{th}\) orbit is \(\frac{1}{{200}}\) times that of the election in the \(n^{th}\) orbit \((C)\) The Ionization energy of muonic atom is \(200\) times more than that of an hydrogen atom \((D)\) The momentum of the muon in the \(n^{th}\) orbit is \(200\) times more than that of the electron