The centre of a wheel rolling on a plane surface moves with a speed \(v_{0} .\) A particle on the rim of the wheel at the same level as the centre will be moving at a speed \(\sqrt{x}\, v_{0}\). Then the value of \(x\) is ...... .

A sample of \(1\) mole gas at temperature \(\mathrm{T}\) is adiabatically expanded to double its volume. If adiabatic constant for the gas is \(\gamma=\frac{3}{2}\), then the work done by the gas in the process is _______.

In the following circuit, the battery has an emf of \(2\mathrm{V}\) and an internal resistance of \(\frac{2}{3}\  \Omega\). The power consumption in the entire circuit is _______ \(W\).

To verify Ohm's law, a student connects the voltmeter across the battery as, shown in the figure. The measured voltage is plotted as a function of the current, and the following graph is obtained If \(V_0\) is almost zero, identify the correct statement

For the four sets of three measured physical quantities as given below. Which of the following options is correct ? \((i)\) \(\mathrm{A}_{1}=24.36, \mathrm{B}_{1}=0.0724, \mathrm{C}_{1}=256.2\) \((ii)\) \(\mathrm{A}_{2}=24.44, \mathrm{B}_{2}=16.082, \mathrm{C}_{2}=240.2\) \((iii)\) \(\mathrm{A}_{3}=25.2, \mathrm{B}_{3}=19.2812, \mathrm{C}_{3}=236.183\) \((iv)\) \(\mathrm{A}_{4}=25, \mathrm{B}_{4}=236.191, \mathrm{C}_{4}=19.5\)

A \(12\,V\) battery connected to a coil of resistance \(6 \,\Omega\) through a switch, drives a constant current in the circuit. The switch is opened in \(1\,ms\). The emf induced across the coil is \(20 \;V\). The inductance of the coil is  ........... \(mH\)

Angular momentum of an electron in a hydrogen atom is \(\dfrac{3h}{\pi}\), then the energy of the electron is _______ eV.

A person of \(80\, kg\) mass is standing on the rim of a circular platform of mass \(200\, kg\) rotating about its axis as \(5\) revolutions per minute \((rpm).\) The person now starts moving towards the centre of the platform. What will be the rotational speed (in \(rpm\) ) of the platform when the person reaches its centre \(......rpm\)

A constant Power delivering machine has towed a box, which was initially at rest, along a horizontal straight line. The distance moved by the box in time \('t'\) is proportional to

A capillary tube made of glass of radius \(0.15\, mm\) is dipped vertically in a beaker filled with methylene iodide (surface tension \(=0.05\, Nm ^{-1},\) density \(\left.=667\, kg m ^{-3}\right)\) which rises to height \(h\) in the tube. It is observed that the two tangents drawn from liquid-glass interfaces (from opp. sides of the capillary) make an angle of \(60^{\circ}\) with one another. Then \(h\) is close to \(......m\) \(\left(g=10\, ms ^{-2}\right)\)

Initially a satellite of 100 kg is in a circular orbit of radius \(1.5 R _{ E }\). This satellite can be moved to a circular orbit of radius \(3 R _{ E }\) by supplying \(\alpha \times 10^6 J\) of energy. The value of \(\alpha\) is _________ .
(Take Radius of Earth \(R_E=6 \times 10^6 m\) and \(g =10 m / s ^2\) )