A particle moves in one dimension from rest under the influence of a force that varies with the distance traveled by the particle as shown in the figure. The kinetic energy of the particle after it has traveled \(3\, m\) is ................ \(\mathrm{J}\)

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of \(\theta ,\) where \(\theta \) is the angle by which it has rotated, is given as \(k\theta ^2.\) If its moment of inertia is \(I\) then the angular acceleration of the disc is

A solid conducting sphere, having a charge \(Q\), is surrounded by an uncharged conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be \(V\). If the shell is now given a charge of \(-4\, Q\), the new potential difference between the same two surface is......\(V\)

For a specific wavelength \(670 \; nm\) of light coming from a galaxy moving with velocity \(v\), the observed wavelength is \(670.7 \; nm\). The value of \(v\) is ............

A child stands on the edge of the cliff \(10\,m\) above the ground and throws a stone horizontally with an initial speed of \(5\,ms ^{-1}\). Neglecting the air resistance, the speed with which the stone hits the ground will be \(..........ms ^{-1}\) (given, \(g =10\,ms ^{-2}\))

The specific heat at constant pressure of a real gas obeying \(\mathrm{PV}^2=\mathrm{RT}\) equation is _______.

The energy of a system is given as \(\mathrm{E}(\mathrm{t})=\alpha^3 \mathrm{e}^{-\beta t}\), where t is the time and \(\beta=0.3 \mathrm{~s}^{-1}\). The errors in the measurement of \(\alpha\) and \(t\) are \(1.2 \%\) and \(1.6 \%\), respectively. At \(t=5 \mathrm{~s}\), maximum percentage error in the energy is :

Two light beams of intensities in the ratio of \(9: 4\) are allowed to interfere. The ratio of the intensity of maxima and minima will be