One main scale division of a vernier caliper is equal to \(m\) units. If \(n^{\text {th }}\) division of main scale coincides with \((n+1)^{\mathrm{th}}\) division of vernier scale, the least count of the vernier caliper is _______.

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length \(L ( R < L )\) about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as M ) :

A coil having \(100\) turns, area of \(5 \times 10^{-3} \mathrm{~m}^2\), carrying current of \(1 \mathrm{~mA}\) is placed in uniform magnetic field of \(0.20 \mathrm{~T}\) such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through \(90^{\circ}\) is _______ \(\mu \mathrm{J}\).

An ice cube of dimensions \(60\,cm \times 50\,cm \times 20\,cm\) is placed in an insulation box of wall thickness \(1\,cm\). The box keeping the ice cube at \(0^{\circ}\,C\) of temperature is brought to a room of temperature \(40^{\circ}\,C\). The rate of melting of ice is approximately. (Latent heat of fusion of ice is \(3.4 \times 10^{5}\,J\,kg ^{-1}\) and thermal conducting of insulation wall is \(0.05\,Wm ^{-10} C ^{-1}\) )

A travelling microscope is used to determine the refractive index of a glass slab. If \(40\) divisions are there in \(1 \; cm\) on main scale and \(50\) Vernier scale divisions are equal to \(49\) main scale divisions, then least count of the travelling microscope is \(\dots \; \times 10^{-6} \; m\)

If the net electric field at point \(\mathrm{P}\) along \(\mathrm{Y}\) axis is zero, then the ratio of \(\left|\frac{q_2}{q_3}\right|\) is \(\frac{8}{5 \sqrt{x}}\), where \(\mathrm{x}=\) _______.

A circular disk of radius R meter and mass M kg is rotating around the axis perpendicular to the disk. An external torque is applied to the disk such that \(\theta(t)=5 t^2-8 t\), where \(\theta(t)\) is the angular position of the rotating disc as a function of time \(t\). How much power is delivered by the applied torque, when \(t=2 s\) ?

A uniform rod \(A B\) of mass \(2 \mathrm{~kg}\) and Length \(30 \mathrm{~cm}\) at rest on a smooth horizontal surface. An impulse of force \(0.2\  \mathrm{Ns}\) is applied to end \(B.\) The time taken by the rod to turn through at right angles will be \(\frac{\pi}{\mathrm{x}}\ \mathrm{s}\), where  X = _______.

The critical angle for a denser-rarer interface is \(45^{\circ}\). The speed of light in rarer medium is \(3 \times 10^8\) ms. The speed of light in the denser medium is:

The velocity \((v)\) and time \((t)\) graph of a body in a straight line motion is shown in the figure. The point \(S\) is at \(4.333\) seconds. The total distance covered by the body in \(6 \;s\) is\(.......m\)

A block of mass \(m\) slides down the plane inclined at angle \(30^{\circ}\) with an acceleration \(\frac{ g }{4}\). The value of coefficient of kinetic friction will be :