A body of mass 2 kg is moving along x -direction such that its displacement as function of time is given by \(x ( t )=\alpha t ^2+\beta t +\gamma m\), where \(\alpha=1 m / s ^2\), \(\beta=1 m / s\) and \(\gamma=1 m\). The work done on the body during the time interval \(t =2 s\) to \(t =3 s\), is _________ J.

Consider two identical springs each of spring constant \(k\) and negligible mass compared to the mass \(M\) as shown. Fig. \(1\) shows one of them and Fig. \(2\) shows their series combination. The ratios of time period of oscillation of the two \(SHM\) is \(\frac{ T _{ b }}{ T _{ a }}=\sqrt{ x },\) where value of \(x\) is (Round off to the Nearest Integer)

If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are \(a, b\) and \(c\) respectively, then the corresponding ratio of increase in their lengths is

Two particles of the same mass are moving in circular orbits because of force, given by \(F(r) = \frac{{ - 16}}{r}\, - \,{r^3}\) The first particle is at a distance \(r = 1,\)  and the second, at \(r = 4.\)  The best estimate for the ratio of kinetic energies of the first and the second particle is closest to

A water spray gun is attached to a hose of cross sectional area \(30\) cm\(^2\). The gun comprises of \(10\) perforations each of cross sectional area \(15\) mm\(^2\). If the water flows in the hose with the speed of \(50\) cm/s, calculate the speed at which the water flows out from each perforation. (Neglect any edge effects)

Proton with kinetic energy of \(1\;MeV\) moves from south to north. It gets an acceleration of \(10^{12}\; \mathrm{m} / \mathrm{s}^{2}\) by an applied magnetic field (west to east). The value of magnetic field :.......\(mT\) (Rest mass of proton is \(1.6 \times 10^{-27} \;\mathrm{kg}\) )

The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of M . Two blocks of mass of M and m are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its centre. The magnitudes of the acceleration experienced by the blocks is _________ (assume no slipping of string on pulley.)

Given below are two statements Statement\(-I :\) Two photons having equal linear momenta have equal wavelengths. Statement\(-II :\) If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease. In the light of the above statements, choose the correct answer from the options given below.

Following gates section is connected in a complete suitable circuit. For which of the following combination, bulb will glow \((ON)\)?

In the given circuit diagram when the current reaches steady state in the circuit, the charge on the capacitor of capacitance \(C\) will be