A vernier callipers has \(20\) divisions on the vernier scale, which coincides with \(19^{\text {th }}\) division on the main scale. The least count of the instrument is \(0.1 \mathrm{~mm}\). One main scale division is equal to _______ \(\mathrm{mm}\).

A particle of mass \(m\) moves on a straight line with its velocity increasing with distance according to the equation \(\mathrm{v}=\alpha \sqrt{\mathrm{x}}\), where \(\alpha\) is a constant. The total work done by all the forces applied on the particle during its displacement from \(\mathrm{x}=0\) to \(\mathrm{x}=\mathrm{d}\), will be _______.

Within a spherical charge distribution of charge density \(\rho \left( r \right)\), \(N\) equipotential surfaces of potential \({V_0},{V_0} + \Delta V,{V_0} + 2\Delta V,\)\(.....{V_0} + N\Delta V\left( {\Delta V > 0} \right),\) are drawn and have increasing radii \(r_0, r_1, r_2,......r_N\), respectively. If the difference in the radii of the surfaces is constant for all values of \(V_0\) and \(\Delta V\) then

Two projectile thrown at \(30^{\circ}\) and \(45^{\circ}\) with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is

The \(K_{\alpha}\; X-\)ray of molybdenum has wavelength \(0.071\, {nm}\). If the energy of a molybdenum atoms with a \(K\) electron knocked out is \(27.5\, {keV}\), the energy of this atom when an \(L\) electron is knocked out will be \(....\,keV.\) (Round off to the nearest integer) \(\left[{h}=4.14 \times 10^{-15} \,{eVs}, {c}=3 \times 10^{8}\, {ms}^{-1}\right.]\)

Two large plane parallel conducting plates are kept 10 cm apart as shown in figure. The potential difference between them is \(V\). The potential difference between the points A and B (shown in the figure) is :

A wire of length \(L\) and radius \(r\) is clamped rigidly at one end. When the other end of the wire is pulled by a force \(f\), its length increases by \(l\). Another wire of same material of length \(2 L\) and radius \(2 r\) is pulled by a force \(2 f\). Then the increase in its length will be

In a screw gauge, there are \(100\) divisions on the circular scale and the main scale moves by \(0.5\,mm\) on a complete rotation of the circular scale. The zero of circular scale lies \(6\) divisions below the line of graduation when two studs are brought in contact with each other. When a wire is placed between the studs, \(4\) linear scale divisions are clearly visible while \(46^{\text {th }}\) division the circular scale coincide with the reference line. The diameter of the wire is \(...........\times 10^{-2}\,mm\)

Three blocks \(A\), \(B\) and \(C\) are pulled on a horizontal smooth surface by a force of \(80 \mathrm{~N}\) as shown in figure. The tensions \(T_1\) and \(T_2\) in the string are respectively _______ and _______.