The magnetic field at the centre of a circular coil of radius \(I\), due to current I flowing through it, is \(B\). The magnetic field at a point along the axis at a distance \(\frac{r}{2}\) from the centre is

The radii of two planets \(A\) and \(B\) are \(R\) and \(4 R\) and their densities are \(\rho\) and \(\rho / 3\) respectively. The ratio of acceleration due to gravity at their surfaces \(\left(g_A: g_B\right)\) will be

Under an adiabatic process, the volume of an ideal gas gets doubled. Consequently the mean collision time between the gas molecule changes from \(\tau_{1}\) to \(\tau_{2} .\) If \(\frac{C_{p}}{C_{v}}=\gamma\) for this gas then a good estimate for \(\frac{\tau_{2}}{\tau_{1}}\) is given by :

A square shaped hole of side \(l=\frac{a}{2}\) is carved out at a distance \(d =\frac{ a }{2}\) from the centre \('O'\) of a uniform circular disk of radius \(a\). If the distance of the centre of mass of the remaining portion from \(O\) is \(-\frac{a}{X},\) value of \(X\) (to the nearest integer) is.......

In a moving coil galvanometer, two moving coils \(M_1\) and \(M_2\) have the following particulars :
\(\begin{aligned}
& \mathrm{R}_1=5 \Omega, \mathrm{~N}_1=15, \mathrm{~A}_1=3.6 \times 10^{-3} \mathrm{~m}^2, \mathrm{~B}_1=0.25 \mathrm{~T} \\
& \mathrm{R}_2=7 \Omega, \mathrm{~N}_2=21, \mathrm{~A}_2=1.8 \times 10^{-3} \mathrm{~m}^2, \mathrm{~B}_2=0.50 \mathrm{~T}
\end{aligned}\)
Assuming that torsional constant of the springs are same for both coils, what will be the ratio of voltage sensitivity of \(M_1\) and \(M_2\) ?

If \(Z=\frac{A^{2} B^{3}}{C^{4}}\), then the relative error in \(Z\) will be

An engine of a train, moving with uniform acceleration, passes the signal-post with velocity \(u\) and the last compartment with velocity \(v\). The velocity with which middle point of the train passes the signal post is

A heavy iron bar, of weight \(\mathrm{W}\) is having its one end on the ground and the other on the shoulder of a person. The bar makes an angle \(\theta\) with the horizontal. The weight experienced by the person is _______.

An massless equilateral triangle \(EFG\) of side \('a'\) (As shown in figure) has three particles of mass \(m\) situated at its vertices. The moment of intertia of the system about the line \(EX\) perpendicular to \(EG\) in the plane of \(EFG\) is \(\frac{ N }{20}\, ma ^{2}\) where \(N\) is an integer. The value of \(N\) is

Two planets have masses \(M\) and \(16\, M\) and their radii are \(a\) and \(2\, a\) respectively. The separation between the centres of the planets is \(10\, a.\) A body of mass \(m\) is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is

Time period of a simple pendulum is \(T\). The time taken to complete \(5 / 8\) oscillations starting from mean position is \(\frac{\alpha}{\beta} T\). The value of \(\alpha\) is ..... .

One end of a straight uniform \(1\; \mathrm{m}\) long bar is pivoted on horizontal table. It is released from rest when it makes an angle \(30^{\circ}\) from the horizontal (see figure). Its angular speed when it hits the table is given as \(\sqrt{\mathrm{n}}\; \mathrm{s}^{-1},\) where \(\mathrm{n}\) is an integer. The value of \(n\) is

A body of mass 2 kg moving with velocity of \(\overrightarrow{\mathrm{v}}_{\mathrm{in}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}} \mathrm{ms}^{-1}\) enters into a constant force field of 6 N directed along positive z -axis. If the body remains in the field for a period of \(\frac{5}{3}\) seconds, then velocity of the body when it emerges from force field is