Two identical strings \(X\) and \(Z\) made of same material have tension \(T _{ x }\) and \(T _{ z }\) in them. If their fundamental frequencies are \(450\, Hz\) and \(300\, Hz ,\) respectively, then the ratio \(T _{ x } / T _{ z }\) is\(.....\)

In a screw gauge the zero of main scale reference line coincides with the fifth division of the circular scale when two studs are in contact. There are \(100\) divisions in circular scale and pitch of screw gauge is \(0.1\) mm. When diameter of a sphere is measured, the reading of main scale is \(5\) mm and \(50^{\text{th}}\) division of circular scale coincides with the reference line of main scale. The diameter of sphere is _______ mm.

Moment of inertia of a rod of mass ' M ' and length 'L' about an axis passing through its center and normal to its length is ' \(\alpha\) '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is :

A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature \(\left(27^{\circ} \mathrm{C}\right)\). The ratio of specific heat of gases at constant volume respectively is _______.

Two particles \(\mathrm{X}\) and \(\mathrm{Y}\) having equal charges are being accelerated through the same potential difference. Thereafter they enter normally in a region of uniform magnetic field and describes circular paths of radius \(R_1\) and \(R_2\) respectively. The mass ratio of \(\mathrm{X}\) and \(\mathrm{Y}\) is _______.

An \(\alpha\) particle and a proton are accelerated from rest by a potential difference of \(200\, V\). After this, their de Broglie wavelengths are \(\lambda_{\alpha}\) and \(\lambda_{p}\) respectively. The ratio \(\frac{\lambda_{p}}{\lambda_{a}}\) is

A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate \(\frac{ dM ( t )}{ dt }= bv ^{2}( t ),\) where \(v ( t )\) is its instantaneous velocity. The instantaneous acceleration of the satellite is

Hydrogen atom is bombarded with electrons accelerated through a potential different of \(\mathrm{V}\), which causes excitation of hydrogen atoms. If the experiment is being formed at \(\mathrm{T}=0 \mathrm{~K}\). The minimum potential difference needed to observe any Balmer series lines in the emission spectra will be \(\frac{\alpha}{10} \mathrm{~V}\), where \(\alpha=\) _______.

Consider a cylinder of mass \(M\) resting on a rough horizontal rug that is pulled out from under it with acceleration \('a'\) perpendicular to the axis of the cylinder. What is \(F_{friction}\) at point \(P\) ? It is assumed that the cylinder does not slip

A particle is projected with velocity \(u\) so that its horizontal range is three times the maximum height attained by it. The horizontal range of the projectile is given as \(\frac{n u^2}{25 g}\), where value of \(n\) is : (Given ' \(g\) ' is the acceleration due to gravity).

A parallel plate capacitor having a separation between the plates \(d\) , plate area \(A\) and material with dielectric constant \(K\) has capacitance \(C_0\). Now one-third of the material is replaced by another material with dielectric constant \(2K\), so that effectively there are two capacitors one with area \(\frac{1}{3}\,A\) , dielectric constant \(2K\) and another with area \(\frac{2}{3}\,A\) and dielectric constant \(K\). If the capacitance of this new capacitor is \(C\) then \(\frac{C}{{{C_0}}}\) is

The root mean square speed of smoke particles of mass \(5 \times 10^{-17}\,kg\) in their Brownian motion in air at NTP is approximately \(.......\,mm\,s ^{-1}\) [Given \(k =1.38 \times 10^{-23}\,J\,K ^{-1}\) ]

In the given figure, the block of mass \(m\) is dropped from the point ' \(A\) '. The expression for kinetic energy of block when it reaches point ' \(B\) ' is ................