If \(50\) Vernier divisions are equal to \(49\) main scale divisions of a travelling microscope and one smallest reading of main scale is \(0.5 \mathrm{~mm}\), the Vernier constant of travelling microscope is _______.

A galvanometer, whose resistance is \(50\,ohm,\) has \(25\) divisions in it. When a current of \(4\times 10^{-4}\,A\) passes through it, its needle (pointer) deflects by one division. To use this galvanometer as a voltmeter of range \(2.5\,V,\) it should be connected to a resistance of.......\(ohm\)

A spherical shell of \(1 \,kg\) mass and radius \(R\) is rolling with angular speed \(\omega\) on horizontal plane (as shown in figure). The magnitude of angular momentum of the shell about the origin \(O\) is \(\frac{a}{3} R^{2} \omega\). The value of a will be ..............

Three identical resistors with resistance \(R=12 \Omega\) and two identical inductors with sell inductance \(L=5 mH\) are connected to an ideal battery with emf of 12 V as shown in figure. The current through the battery long after the switch has been closed will be _________ A.

A small bob tied at one end of a thin string of length \(1\, m\) is describing a vertical circle so that the maximum and minimum tension in the string are in the ratio \(5: 1 .\) The velocity of the bob at the height position is ............. \(m / s\) (Take \(\left.g=10 \,m / s ^{2}\right)\)

An aluminium rod with Young's modulus \(Y =7.0\) \(\times 10^{10} N / m ^2\) undergoes elastic strain of \(0.04 \%\). The energy per unit volume stored in the rod in SI unit is:

A potentiometer wire \(PQ\) of \(1\,m\) length is connected to a standard cell \(E _{1}\). Another cell \(E _{2}\) of emf \(1.02\, V\) is connected with a resistance \('r'\) and switch \(S\) (as shown in figure). With switch \(S\) open, the null position is obtained at a distance of \(49\, cm\) from \(Q\). The potential gradient in the potentiometer wire is.......\(V/cm\)

One mole of a diatomic ideal gas undergoes a cyclic process \(ABC\) as shown in figure. The process \(BC\) is adiabatic. The temperatures at \(A, B\) and \(C\) are \(400\ K, 800\ K \) and \(600\ K\) respectively. Choose the correct statement

Electric field of plane electromagnetic wave propagating through a non-magnetic medium is given by \({E}=20 \cos \left(2 \times 10^{10} {t}-200 {x}\right) \,{V} / {m} .\) The dielectric constant of the medium is equal to : (Take \(\mu_{{r}}=1\) )

Moment of inertia of a cylinder of mass \(M,\) length \(L\) and radius \(R\) about an axis passing through its centre and perpendicular to the axis of the cylinder is \(I = M \left(\frac{ R ^{2}}{4}+\frac{ L ^{2}}{12}\right) .\) If such a cylinder is to be made for a given mass of material, the ratio \(\frac LR\) for it to have minimum possible \(I\) is