The moment of inertia of a circular ring of mass \(M\) and diameter r about a tangential axis lying in the plane of the ring is :

A body of mass ' \(m\) ' is projected with a speed ' \(u\) ' making an angle of \(45^{\circ}\) with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as \(\frac{\sqrt{2} \mathrm{mu}^3}{\mathrm{Xg}}\). The value of ' \(\mathrm{X}\) ' is _______.

An EM wave from air enters a medium. The electric fields are \(\overrightarrow {{E_1}}  = {E_{01}}\hat x\;cos\left[ {2\pi v\left( {\frac{z}{c} - t} \right)} \right]\) in air and \(\overrightarrow {{E_2}}  = {E_{02}}\hat x\;cos\left[ {k\left( {2z - ct} \right)} \right]\) in medium, where the wave number \(k\) and frequency \(v\) refer to their values in air. The medium is nonmagnetic. If \(\varepsilon {_{{r_1}}}\) and \(\varepsilon {_{{r_2}}}\) refer to relative permittivities of air and medium respectively, which of the following options is correct?

In a Young's double slit experiment, the intensities at two points, for the path difference \(\frac{\lambda}{4}\) and \(\frac{\lambda}{3}\) ( \(\lambda\) being the wavelength of light used) are \(I_1\) and \(I_2\) respectively. If \(I_0\) denotes the intensity produced by each one of the individual slits, then \(\frac{ I _1+ I _2}{ I _0}=..............\)

A particle of mass \(m\) is acted upon by a force \(F\) given by the empirical law \(F = \frac{R}{{{t^2}}}\,v(t)\). If this law is to be tested experimentally by observing the motion starting from rest, the best way is to plot 

In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its \(4^{\text {th }}\) division coincides exactly with a certain division on main scale. If \(50\) vernier scale divisions equal to \(49\) main scale divisions and zero error in the instrument is \(0.04 \mathrm{~mm}\) then how many main scale divisions are there in \(1 \mathrm{~cm}\) ?

The length of a light string is 1.4 m when the tension on it is 5 N. If the tension increases to 7 N , the length of the string is 1.56 m. The original length of the string is _____ m.

In an experiment, a small steel ball falls through a Iiquid at a constant speed of \(10\, cm/s\). If the steel ball is pulled upward with a force equal to twice its effective weight, how fast will it move upward ? ......... \(cm/s\)

A solid body of constant heat capacity \(1\ J/^o C\) is being heated by keeping it in contact with reservoirs in two ways : \((i)\) Sequentially keeping in contact with \(2\) reservoirs such that each reservoir supplies same amount of heat. \((ii)\) Sequentially keeping in contact with \(8\) reservoirs such that each reservoir supplies same amount of heat. In both the cases body is brought from initial temperature \(100^o C\) to final temperature \(200^o C\). Entropy change of the body in the two cases respectively is :

Two planets \(A\) and \(B\) of radii \(R\) and \(1.5 R\) have densities \(\rho\) and \(\rho / 2\) respectively. The ratio of acceleration due to gravity at the surface of \(B\) to \(A\) is :

A wire of density \(8 \times 10^3\,kg / m ^3\) is stretched between two clamps \(0.5\,m\) apart. The extension developed in the wire is \(3.2 \times 10^{-4}\,m\). If \(Y =8 \times 10^{10}\,N / m ^2\), the fundamental frequency of vibration in the wire will be \(......\,Hz\).