The mass of the moon is \(1 / 144\) times the mass of a planet and its diameter \(1 / 16\) times the diameter of a planet. If the escape velocity on the planet is \(\mathrm{v}\), the escape velocity on the moon will be _______.

A cylindrical rod of length 1 m and radius 4 cm is mounted vertically. It is subjected to a shear force of \(10^5 \mathrm{~N}\) at the top. Considering infinitesimally small displacement in the upper edge, the angular displacement \(\theta\) of the rod axis from its original position would be : (shear moduli, \(\left.\mathrm{G}=10^{10} \mathrm{~N} / \mathrm{m}^2\right)\)

In a Young's double slit experiment \(15\) fringes are observed on a small portion of the screen when light of wavelength \(500 \;\mathrm{nm}\) is used. Ten fringes are observed on the same section of the screen when another light source of wavelength \(\lambda\) is used. Then the value of \(\lambda\) is (in \(\mathrm{nm})\)

\(M\) grams of steam at \(100^{\circ} \mathrm{C}\) is mixed with \(200\; \mathrm{g}\) of ice at its melting point in a thermally insulated container. If it produces liquid water at \(40^{\circ} \mathrm{C}\) [heat of vaporization of water is \(540 \;cal/\mathrm{g}\) and heat of fusion of ice is \(80 \;\text { cal/g }]\) the value of \(\mathrm{M}\) is

A current of \(5\, A\) passes through a copper conductor (resistivity \(= 1.7\times10^{-8}\,\Omega \,m\)) of radius of cross-section \(5\, mm\). Find the mobility of the charges if their drift velocity is \(1.1\times10^{-3}\, m/s\) ................ \(m^2/Vs\)

In Young's double slit experiment, the distance between slits and the screen is \(1 .0\,\,m\) and monochromatic light of \(600\,\,nm\)  is being used. A person standing near the slits is looking at the fringe pattern. When the separation between the slits is varied, the interference pattern disappears for a particular distance \(d_0\) between the slits . If the angular resolution of the eye is \(\frac {1}{60}^o,\) the value of \(d_0\)  is close to......\(mm\)

If one mole of an ideal gas at \(\left( P _{1}, V _{1}\right)\) is allowed to expand reversibly and isothermally (\(A\) to \(B\) ) its pressure is reduced to one-half of the original pressure (see \(figure\)). This is followed by a constant volume cooling till its pressure is reduced to one-fourth of the initial value \(( B \rightarrow C ) .\) Then it is restored to its initial state by a reversible adiabatic compression (\(C\) to \(A\)). The net workdone by the gas is equal to  ...... .

A bullet of \('4\,g'\) mass is fired from a gun of mass \(4 \,{kg}\). If the bullet moves with the muzzle speed of \(50\, {ms}^{-1}\), the impulse imparted to the gun and velocity of recoil of gun are :

\(\left(P+\frac{a}{V^2}\right)(V-b)=R T\) represents the equation of state of some gases. Where \(P\) is the pressure, \(V\) is the volume, \(T\) is the temperature and \(a, b, R\) are the constants. The physical quantity, which has dimensional formula as that of \(\frac{b^2}{a}\), will be

A nucleus \(A\), with a finite de -broglie wavelength \(\lambda_A\), undergoes spontaneous fission into two nuclei \(B\) and \(C\) of equal mass. \(B\) flies in the same direction as that of \(A\), while \(C\) flies in the opposite direction with a velocity equal to half of that of \(B\). The de -Broglie wavelength \(\lambda_B\) and \(\lambda_C\) of \(B\) and \(C\) are respectively

Given below are two statements : Statement \((I)\) : dimensions of specific heat is \(\left[\mathrm{L}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]\) Statement \((II)\) : dimensions of gas constant is \(\left[\mathrm{ML}^2 \mathrm{~T}^{-1} \mathrm{~K}^{-1}\right]\)

A particle of charge \(q\) and mass \(m\) is moving with a velocity \(-v \hat{ i }(v \neq 0)\) towards a large screen placed in the \(Y - Z\) plane at a distance \(d.\) If there is a magnetic field \(\overrightarrow{ B }= B _{0} \hat{ k },\) the minimum value of \(v\) for which the particle will not hit the screen is