When a particle of mass \(m\) is attached to a vertical spring of spring constant \(k\) and released, its motion is described by \(y ( t )= y _{0} \sin ^{2} \omega t ,\) where \('y'\) is measured from the lower end of unstretched spring. Then \(\omega\) is

At a given instant, say \(t = 0,\) two radioactive substances \(A\) and \(B\) have equal activates. The ratio \(\frac{{{R_B}}}{{{R_A}}}\) of their activities. The ratio \(\frac{{{R_B}}}{{{R_A}}}\) of their activates after time \(t\) itself decays with time \(t\) as \(e^{-3t}.\) If the half-life of \(A\) is \(ln2,\) the half-life of \(B\) is

The time period of a satellite in a circular orbit of radius \(R\) is \(T\). The period of another satellite in a circular orbit of radius \(9 R\) is............ \(T\)

Two slabs with square cross section of different materials \((1,2)\) with equal sides \((l)\) and thickness \(\mathrm{d}_1\) and \(\mathrm{d}_2\) such that \(\mathrm{d}_2=2 \mathrm{~d}_1\) and \(l \gt \mathrm{d}_2\). Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of deformation is \(\theta_2=2 \theta_1\). If the shear moduli of material 1 is \(4 \times 10^9 \mathrm{~N} / \mathrm{m}^2\), then shear moduli of material 2 is \(\mathrm{x} \times 10^9 \mathrm{~N} / \mathrm{m}^2\), where value of \(x\) is ________.

Find out the surface charge density at the intersection of point \(x =3\, m\) plane and \(x\) -axis, in the region of uniform line charge of \(8\, nC / m\) lying along the \(z\) -axis in free space.

Two charges of \(-4 \ \mu \mathrm{C}\) and \(+4\  \mu \mathrm{C}\) are placed at the points \(A(1,0,4) \mathrm{m}\) and \(B(2,-1,5) \mathrm{m}\) located in an electric field \(\vec{E}=0.20 \hat{\mathrm{i}} \mathrm{V} / \mathrm{cm}\). The magnitude of the torque acting on the dipole is \(8 \sqrt{\alpha} \times 10^{-3} \mathrm{Nm}\), Where \(\alpha=\)___________

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

Two ions of masses \(4 \,{amu}\) and \(16\, amu\) have charges \(+2 {e}\) and \(+3 {e}\) respectively. These ions pass through the region of constant perpendicular magnetic field. The kinetic energy of both ions is same. Then :

A diatomic gas \((\gamma=1.4)\) does \(200 \mathrm{~J}\) of work when it is expanded isobarically. The heat given to the gas in the process is _______.

One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is \(\frac{\alpha^{2}}{4} R J / mol\,K\); then the value of \(\alpha\) will be \(.......\) (Assume that the given diatomic gas has no vibrational mode.)

When a hydrogen atom going from \(\mathrm{n}=2\) to \(\mathrm{n}=1\) emits a photon, its recoil speed is \(\frac{\mathrm{x}}{5} \mathrm{~m} / \mathrm{s}\). Where \(\mathrm{x}=\) _______. (Use : mass of hydrogen atom\(\left.=1.6 \times 10^{-27 /} \ \mathrm{kg}\right)\)