A sphere of mass \(2\,kg\) and radius \(0.5\, m\) is rolling with an initial speed of \(1 \,ms ^{-1}\) goes up an inclined plane which makes an angle of \(30^{\circ}\) with the horizontal plane, without slipping. How low will the sphere take to return to the starting point \(A\) ? (in \(second\))

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of \(\mathrm{M}^{\mathrm{P}} \mathrm{L}^{\mathrm{Q}} \mathrm{T}^{\mathrm{R}} \mathrm{A}^{\mathrm{S}}\), where value of ' Q ' and ' R ' are

If the acceleration due to gravity experienced by a point mass at a height \(h\) above the surface of earth is same as that of the acceleration due to gravity at a depth a depth \(\alpha\) h \(\left(h \ll R_{e}\right)\) from the earth surface. The value of \(\alpha\) will be\(....\)(use \(R _{ e }=6400\,km\) )

Two masses m and 2m are connected by a light string going over a pulley (disc) of mass 30m with radius r = 0.1 m. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The 2m mass is released from rest and its speed when it has descended through a height of 3.6 m is __________ m/s. (Assume string does not slip and \( g=10 m/s^{2} \))

Given below are two statements Statement \(-I\) : A point charge is brought in an electric field. The value of electric field at a point near to the charge may increase if the charge is positive. Statement \(-II\) : An electric dipole is placed in a non-uniform electric field. The net electric force on the dipole will not be zero. Choose the correct answer from the options given below 

A ball projected from ground at an angle of \(45^o\) just clears a wall in front. If point of projection is \(4\,m\) from the foot of wall and ball strikes the ground at a distance of \(6\,m\) on the other side of the wall, the height of the walI is  ........ \( m\)

An \(npn\) transistor operates as a common emitter amplifier, with a power gain of \(60\, dB\). The input circuit resistance Is \(100\,\Omega \) and the output load resistance is \(10 \,k\,\Omega \). the common emitter current gain \(\beta \) is

A small particle moves to position \(5 \hat{i}-2 \hat{j}+\hat{k}\) from its initial position \(2 \hat{i}+3 \hat{j}-4 \hat{k}\) under the action of force \(5 \hat{i}+2 \hat{j}+7 \hat{k} N\). The value of work done will be \(............J\).

Two nuclei of mass number \(3\) combine with another nucleus of mass number \(4\) to yield a nucleus of mass number \(10\). If the binding energy per nucleon for the mass numbers \(3\), \(4\) and \(10\) are \(5.6\) MeV, \(7.4\) MeV and \(6.1\) MeV, respectively, then in the process, \(\Delta Mc^2 = \) _____ MeV.