A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights \(h_{sph}\) and \(h_{cyl}\) on the incline. The radio \(\frac{{{h_{sph}}}}{{{h_{cyl}}}}\) is given by

A small rigid spherical ball of mass M is dropped in a long vertical tube containing glycerine. The velocity of the ball becomes constant after some time. If the density of glycerine is half of the density of the ball, then the viscous force acting on the ball will be (consider g as acceleration due to gravity)

A sonometer wire of resonating length \(90 \mathrm{~cm}\) has a fundamental frequency of \(400 \mathrm{~Hz}\) when kept under some tension. The resonating length of the wire with fundamental frequency of \(600 \mathrm{~Hz}\) under same tension ________ \(\mathrm{cm}\).

The volume of an ideal gas \((\gamma=1.5)\) is changed adiabatically from \(5\) litres to \(4\) litres. The ratio of initial pressure to final pressure is _______.

A circular disc reaches from top to bottom of an inclined plane of length \(l\). When it slips down the plane, if takes \(\mathrm{t} \mathrm{s}\). When it rolls down the plane then it takes \(\left(\frac{\alpha}{2}\right)^{1 / 2} \mathrm{t} s\), where \(\alpha\) is. _______.

Consider a sphere of radius \(\mathrm{R}\) which carries a uniform charge density \(\rho .\) If a sphere of radius \(\frac{\mathrm{R}}{2}\) is carved out of it, as shown, the ratio \(\frac{\left|\overrightarrow{\mathrm{E}}_{\mathrm{A}}\right|}{\left|\overrightarrow{\mathrm{E}}_{\mathrm{B}}\right|}\) of magnitude of electric field \(\overrightarrow{\mathrm{E}}_{\mathrm{A}}\) and \(\overrightarrow{\mathrm{E}}_{\mathrm{B}}\) respectively, at points \(\mathrm{A}\) and \(\mathrm{B}\) due to the remaining portion is

Given below are two statements: Statement \((I)\) : When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account. Statement \((II)\) : Ampere's circuital law does not depend on Biot-Savart's law. In the light of the above statements, choose the correct answer from the options given below:

Each of three blocks \(P\), \(Q\) and \(R\) shown in figure has a mass of \(3 \mathrm{~kg}\). Each of the wire \(A\) and \(B\) has cross-sectional area \(0.005 \mathrm{~cm}^2\) and Young's modulus \(2 \times 10^{11} \mathrm{~N} \mathrm{~m}^{-2}\). Neglecting friction, the longitudinal strain on wire \(B\) is _______ \(\times 10^{-4}\). \(\left(\right.\) Take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) )

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 \(1.6 \times 10^{-27} \;\mathrm{kg}\) )

A block of mass \(5 \mathrm{~kg}\) is placed on a rough inclined surface as shown in the figure.If \(\vec{F}_1\) is the force required to just move the block up the inclined plane and \(\vec{F}_2\) is the force required to just prevent the block from sliding down, then the value of \(\left|\vec{F}_1\right|-\left|\vec{F}_2\right|\) is _______. [Use \(g=10 \mathrm{~m} / \mathrm{s}^2\) ]