A \(1 \,kg\) block attached to a spring vibrates with a frequency of \(1\, Hz\) on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an \(8\, kg\) block placed on the same table. So, the frequency of vibration of the \(8\, kg\) block is ..... \(Hz\)

The pair of physical quantities not having same dimensions is :

A body of mass 14 kg initially at rest explodes and breaks into three fragments of masses in the ratio \(2:2:3\). The two pieces of equal masses fly off perpendicular to each other with a speed of \(18~m/s\) each. The velocity of the heavier fragment is ___________ \(m/s\).

The charge accumulated on the capacitor connected in the following circuit is________ \(\mu \mathrm{C}\) (Given \(\mathrm{C}=150\  \mu \mathrm{F}\))

Two discs of moment of inertia \(I_1=4 \mathrm{~kg} \mathrm{~m}^2\) and \(\mathrm{I}_2=2 \mathrm{~kg} \mathrm{~m}^2\) about their central axes & normal to their planes, rotating with angular speeds \(10 \mathrm\ {rad} / \mathrm{s}\) & \(4  \mathrm\ {rad} / \mathrm{s}\) respectively are brought into contact face to face with their axe of rotation coincident. The loss in kinetic energy of the system in the process is _______ \(\mathrm{J}\).

In a Young's double slit experiment with light of wavelength \(\lambda \) the separation of slits is \(d\) and distance of screen is \(D\) such that \(D >> d >> \lambda \). If the fringe width is \(\beta \). the distance from point of maximum intensity to the point where intensity falls to half of maximum intensity on either side is

Monochromatic light of wavelength \(500 \mathrm{~nm}\) is used in Young's double slit experiment. An interference pattern is obtained on a screen When one of the slits is covered with a very thin glass plate (refractive index \(=1.5\) ), the central maximum is shifted to a position previously occupied by the \(4^{\text {th }}\) bright fringe. The thickness of the glass-plate is _______\(\mu \mathrm{m}\).

A particle of mass \(1 \,{mg}\) and charge \(q\) is lying at the mid-point of two stationary particles kept at a distance \('2 \,{m}^{\prime}\) when each is carrying same charge \('q'.\) If the free charged particle is displaced from its equilibrium position through distance \('x'\) \((x\,< \,1\, {m})\). The particle executes \(SHM.\) Its angular frequency of oscillation will be \(....\,\times 10^{8}\, {rad} / {s}\) if \({q}^{2}=10\, {C}^{2}\)

A mass of \(1\text{ kg}\) is kept on an inclined plane with \(30°\) inclination with respect to horizontal plane and it is at rest initially. Then the whole assembly is moved up with constant velocity of \(4\text{ m/s}\). The work done by the frictional force in time \(2\text{ s}\) is _______ J. (Take \(g = 10\text{ m/s}^2\))

A certain gas is isothermally compressed to \(\left(\dfrac{1}{3}\right)^{rd}\) of its initial volume (\(V_o = 3\) litre) by applying required pressure. If the bulk modulus of the gas is \(3 \times 10^5\) N/m\(^2\), the magnitude of work done on the gas is _______ J.

The circuit shown below contains two ideal diodes, each with a forward resistance of \(50\,\Omega \). If the battery voltage is \(6\,V\), the current through the \(100\,\Omega \) resistance (in Amperes) is