Two satellites, \(A\) and \(B,\) have masses \(m\) and \(2\,m\) respectively. \(A\) is in a circular orbit of radius \(R,\) and \(B\) is in a circular orbit of radius \(2R\) around the earth. The ratio of their kinetic energies, \(K.E._A / K.E._B ,\) is

A sample of an ideal gas is taken through the cyclic process \(abca\) as shown in the figure. The change in the internal energy of the gas along the path \(ca\) is \(-180\, J\). The gas absorbs \(250\, J\) of heat along the path \(ab\) and \(60\, J\) along the path \(bc\). The work done by the gas along the path \(abc\) is ..... \(J\)

From the \(v\) - \(t\) graph shown. the ratio of distance to displacement in \(25\,s\) of motion

Two identical blocks \(A\) and \(B\) each of mass \(m\) resting on the smooth horizontal floor are connected by a light spring of natural length \(L\) and spring constant \(K\). A third block \(C\) of mass \(m\) moving with a speed \(v\) along the line joining \(A\) and \(B\) collides with \(A\).The maximum compression in the spring is

A proton with a kinetic energy of \(2.0\,eV\) moves into a region of uniform magnetic field of magnitude \(\frac{\pi}{2} \times 10^{-3}\,T\). The angle between the direction of magnetic field and velocity of proton is \(60^{\circ}\). The pitch of the helical path taken by the proton is \(..........cm\) (Take, mass of proton \(=1.6 \times 10^{-27}\,kg\) and Charge on proton \(=1.6 \times 10^{-19}\,kg)\)

A spring whose unstretched length is \(\ell \) has a force constant \(k\). The spring is cut into two pieces of unstretched lengths \(\ell_1\) and \(\ell_2\) where, \(\ell_1 = n\ell_2\) and \(n\) is an integer. The ratio \(k_1/k_2\) of the corresponding force constants, \(k_1\) and \(k_2\) will be

Let \(N_{\beta}\) be the number of \(\beta \) particles emitted by \(1\) gram of \(Na^{24}\) radioactive nucler (half life \(= 15\, hrs\)) in \(7.5\, hours\), \(N_{\beta}\) is close to (Avogadro number \(= 6.023\times10^{23}\,/g.\, mole\))

The length of a metallic wire is increased by \(20 \%\) and its area of cross section is reduced by \(4 \%\). The percentage change in resistance of the metallic wire is \(.......\)

In a plane electromagnetic wave, the directions of electric field and magnetic field are represented by \(\hat{ k }\) and \(2 \hat{ i }-2 \hat{ j },\) respectively What is the unit vector along direction of propagation of the wave.

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal place: \((i)\) a ring of radius \(R,\,(ii)\) a solid cylinder of radius \(\frac {R}{2}\)  nd \((iii)\) a solid sphere of radius \(\frac {R}{4}.\) If, in each case, the speed of the center of mass at bottom of the incline is same, the ratio of the maximum heights they climb is