Three balls of masses \(2 \mathrm{~kg}, 4 \mathrm{~kg}\) and \(6 \mathrm{~kg}\) respectively are arranged at centre of the edges of an equilateral triangle of side \(2 \mathrm{~m}\). The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be _______\(\mathrm{kg} \mathrm{m}{ }^2\).

At some instant, a radioactive sample \(S_1\) having an activity \(5\,\mu Ci\)  has twice the number of nuclei as another sample \(S_2\)  which has an activity of  \(10\,\mu Ci.\)  The halflives of \(S_1\)  and  \(S_2\) are

As shown in the figure, forces of \(10^5\,N\) each are applied in opposite directions, on the upper and lower faces of a cube of side \(10\,cm\), shifting the upper face parallel to itself by \(0.5\,cm\) . If the side of another cube of the same material is, \(20\,cm\) then under similar conditions as above, the displacement will be......... \(cm\)

A plane electromagnetic wave of wavelength \(\lambda \) has an intensity \(I.\)  It is propagating along the positive \(Y-\)  direction. The allowed expressions for the electric and magnetic fields are given by

In the figure, given that \(V_{BB}\) supply can vary from \(0\) to \(5.0 \,V,\,\,V_{CC} = 5\,V,\) \(\beta _{dc} = 200,\,\,R_B = 100,\,k\Omega ,\) \(R_C = 1\,k\Omega  \) an \(V_{BE} = 1.0\,V,\) The minimum base current and the input voltage at which the transistor will go to saturation, will be respectively

Some energy levels of a molecule are shown in the figure. The ratio of the wavelengths \(r\) =\(\frac{{{\lambda _1}}}{{{\lambda _2}}}\) is given by

As shown in figure, a \(70\,kg\) garden roller is pushed with a force of \(\overrightarrow{ F }=200\,N\) at an angle of \(30^{\circ}\) with horizontal. The normal reaction on the roller is \(.......\,N\) \(\left(\right.\) Given \(\left.g =10\,m s ^{-2}\right)\)

A power transmission line feeds input power at \(2300\,V\) to a step down transfonner with its primary windings having  \(4000\) turns, giving the output power at \(230\,V.\)  If the current in the primary of the transformer is  \(5\,A,\) and its efficiency is \(90\%,\)  the output current would be......\(A\)

The escape velocities of two planets \(A\) and \(B\) are in the ratio \(1: 2\). If the ratio of their radii respectively is \(1: 3\), then the ratio of acceleration due to gravity of planet \(A\) to the acceleration of gravity of planet \(B\) will be:

Three different processes that can occur in an ideal monoatomic gas are shown in the \(P\) vs \(V\) diagram. The paths are labelled as \(A \rightarrow B, A \rightarrow C\) and \(A \rightarrow D .\) The change in internal energies during these process are taken as \(E _{ AB }, E _{ AC }\) and \(E _{ AD }\) and the work done as \(W _{ AB }\) \(W _{ AC }\) and \(W _{ AD }\) The correct relation between these parameters are