The radius in kilometer to which the present radius of earth \(( R =6400\, km )\) to be compressed so that the escape velocity is increased \(10\) time is ............\(km\)

Two stars are \(10\) light years away from the earth. They are seen through a telescope of objective diameter \(30\, cm\). The wavelength of light is \(600\, nm\). To see the stars just resolved by the telescope, the minimum distance between them should be ( \(1\) light year \(= 9.46 \times 10^{15}\, m\)) of the order of

The magnifying power of a telescope with tube \(60\; \mathrm{cm}\) is \(5 .\) What is the focal length of its eye piece?......\(cm\)

A vector has magnitude same as that of \(\overrightarrow{\mathrm{A}}-=3 \hat{\mathrm{j}}+4 \hat{\mathrm{j}}\) and is parallel to \(\overrightarrow{\mathrm{B}}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}\). The \(\mathrm{x}\) and \(y\) components of this vector in first quadrant are \(\mathrm{x}\) and \(3\) respectively where  \(x\)= _______.

A disc is rolling without slipping on a surface. The radius of the disc is \(R\). At \(t=0\), the top most point on the disc is \(A\) as shown in figure. When the disc completes half of its rotation, the displacement of point \(A\) from its initial position is

The magnetic susceptibility of a material of a rod is \(499 .\) Permeability in vaccum is \(4 \pi \times 10^{-7}\; {H} / {m}\). Absolute permeability of the material of the rod is \(....\,\pi \times 10^{-4} {H} / {m}\)

The terminal velocity \(\left( v _{ t }\right)\) of the spherical rain drop depends on the radius ( \(r\) ) of the spherical rain drop as

A particle of mass \(500 \,gm\) is moving in a straight line with velocity \(v=b x^{5 / 2}\). The work done by the net force during its displacement from \(x=0\) to \(x =4 \,m\) is ...................\(J\) (Take \(b =0.25 \,m ^{-3 / 2} s ^{-1}\) )

Three point charges \(q,-2 q\) and \(2 q\) are placed on \(x\)-axis at a distance \(x=0, x=\frac{3}{4} R\) and \(x=R\) respectively from origin as shown. If \(q =2 \times 10^{-6}\,C\) and \(R =2\,cm\), the magnitude of net force experienced by the charge \(-2 q\) is .......... \(N\)

Consider two thin identical conducting wires covered with very thin insulating material. One of the wires is bent into a loop and produces magnetic field \(B_1,\) at its centre when a current \(I\) passes through it.The second wire is bent into a coil with  three identical loops adjacent to each other and produces magnetic field \(B_2\) at the centre of the loops when current \(I/3\) passes through it. The ratio \(B_1 : B_2\) is

Given below is the plot of a potential energy function \(U(x)\) for a system, in which a particle is in one dimensional motion, while a conservative force \(F(x)\) acts on it. Suppose that \(E_{\text {mech }}=8\, {J}\), the incorrect statement for this system is: