A bead of mass \(m\) stays at point \(P ( a , b )\) on a wire bent in the shape of a parabola \(y=4 Cx ^{2}\) and rotating with angular speed \(\omega\) (see figure). The value of \(\omega\) is (neglect friction)

Water drops are falling from a nozzle of a shower onto the floor, from a height of \(9.8\, {m}\). The drops fall at a regular interval of time. When the first drop strikes the floor, at that instant, the third drop begins to fall. Locate the position of second drop from the floor when the first drop strikes the floor.  (In \(m\))

A bag of sand of mass \(9.8\,kg\) is suspended by a rope. A bullet of \(200 \,g\) travelling with speed \(10\, ms ^{-1}\) gets embedded in it, then loss of kinetic energy will be \(...J\)

The refractive index of the material of a glass prism is \(\sqrt{3}\). The angle of minimum deviation is equal to the angle of the prism. What is the angle of the prism?

A vector \(\vec A \) is rotated by a small angle \(\Delta \theta\) radian \(( \Delta \theta << 1)\) to get a new vector \(\vec B\) In that case \(\left| {\vec B - \vec A} \right|\) is

Which of the following equations is dimensionally incorrect? Where \(t=\) time, \(h=\) height, \(s=\) surface tension, \(\theta=\) angle, \(\rho=\) density, \(a, r=\) radius, \(g=\) acceleration due to gravity, \({v}=\) volume, \({p}=\) pressure, \({W}=\) work done, \(\Gamma=\) torque, \(\varepsilon=\) permittivity, \({E}=\) electric field, \({J}=\) current density, \({L}=\) length.

Suppose a uniformly charged wall provides a uniform electric field of \(2 \times 10^4 \mathrm{~N} / \mathrm{C}\) normally. A charged particle of mass \(2 \mathrm{~g}\) being suspended through a silk thread of length \(20 \mathrm{~cm}\) and remain stayed at a distance of \(10 \mathrm{~cm}\) from the wall. Then the charge on the particle will be \(\frac{1}{\sqrt{\mathrm{x}}} \ \mu \mathrm{C}\) where \(\mathrm{x}=\) _______ [use \(g=10 \mathrm{~m} / \mathrm{s}^2\)]

The ratio of momentum of the photons of the \(1^{st}\) and \(2^{nd}\) line of Balmer series of Hydrogen atoms is \(\alpha/\beta\). The possible values of \(\alpha\) and \(\beta\) are:-

For the given logic gate circuit, which of the following is the correct truth table?

Consider two uniform discs of the same thickness and different radii \(R _{1}= R\) and \(R _{2}=\alpha R\) made of the same material. If the ratio of their moments of inertia \(I_{1}\) and \(I_{2}\), respectively, about their axes is \(I _{1}: I _{2}=1: 16\) then the value of \(\alpha\) is

Two coaxial solenoids of different radii carry current \(I\) in the same direction. Let \(\;{\overrightarrow {\;F} _1}\) be the magnetic force on the inner solenoid due to the outer one and \(\;{\overrightarrow {\;F} _2}\) be the magnetic force on the outer solenoid due to the inner one. Then

If earth has a mass nine times and radius twice to the of a planet \(P\). Then \(\frac{v_e}{3} \sqrt{x}\; ms ^{-1}\) will be the minimum velocity required by a rocket to pull out of gravitational force of \(P\), where \(v_e\) is escape velocity on earth. The value of \(x\) is

A helicopter flying horizontally with a speed of \(360 \mathrm{~km} / \mathrm{h}\) at an altitude of 2 km , drops an object at an instant. The object hits the ground at a point O , 20 s after it is dropped. Displacement of ' O ' from the position of helicopter where the object was released is :
(use acceleration due to gravity \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) and neglect air resistance)