The de-Broglie wavelength associated with a particle of mass \(m\) and energy \(E\) is \(\mathrm{h} / \sqrt{2 m E}\) The dimensional formula for Planck's constant is _______.

Modern vacuum pumps can evacuate a vessel down to a pressure of \(4.0 \times {10^{ - 15}}\, atm\) at room temperature \((300\, K)\). Taking \(R = 8.0\, JK^{-1}\, mole^{-1}\) , \(1\, atm = 10^5\, Pa\) and \(N_ {Avogadro} = 6 \times 10^{23}\, mole^{-1}\) , the mean distance between molecules of gas in an evacuated vessel will be of the order of

A person moved from \(A\) to \(B\) on a circular path as shown in figure. If the distance travelled by him is \(60 \,m\), then the magnitude of displacement would be\(.....\,m\) (Given \(\left.\cos 135^{\circ}=-0.7\right)\)

A bullet of mass \(10\, g\) and speed \(500\, m/s\) is fired into a door and gets embedded exactly at the centre of the door. The door is \(1.0\, m\) wide and weighs \(12\, kg\). It is hinged at one end and rotates about a vertical axis practically without friction . The angular speed of the door just after the bullet embeds into it will be

A bowl filled with very hot soup cools from \(98^{\circ}\,C\) to \(86^{\circ}\,C\) in \(2\) minutes when the room temperature is \(22^{\circ}\,C\). \(..........\,minutes\) long it will take to cool from \(75^{\circ}\,C\) to \(69^{\circ}\,C\) ?

For a plane electromagnetic wave, the magnetic field at a point \(x\) and time \(t\) is \(\overrightarrow{ B }( x , t )=\left[1.2 \times 10^{-7} \sin \left(0.5 \times 10^{3} x +1.5 \times 10^{11} t \right) \hat{ k }\right] T\) The instantaneous electric field \(\overrightarrow{ E }\) corresponding to \(\overrightarrow{ B }\) is : (speed of light \(\left.c=3 \times 10^{8} ms ^{-1}\right)\)

Which of the following statements is false?

A horse rider covers half the distance with \(5\,m / s\) speed. The remaining part of the distance was travelled with speed \(10\,m / s\) for half the time and with speed \(15 m / s\) for other half of the time. The mean speed of the rider averaged over the whole time of motion is \(\frac{x}{7}\,m / s\). The value of \(x\) is

A tennis ball (treated as hollow spherical shell) starting from \(O\) rolls down a hill. At point \(A\) the ball becomes air borne leaving at an angle of \(30^o\) with the horizontal. The ball strikes the ground at \(B\). What is the value of the distance \(AB\) ? (Moment of inertia of a spherical shell of mass \(m\) and radius \(R\) about its diameter .......... \(m\). \(= \frac {2}{3}\,mR^2\))

A spring mass system (mass \(m\), spring constant \(\mathrm{k}\) and natural length \(l\) ) rest in equilibrium on a horizontal disc. The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system, rotates about it's axis with an angular velocity \(\omega,\left(\mathrm{k} > > \mathrm{m} \omega^{2}\right)\) the relative change in the length of the spring is best given by the option

Consider a cylindrical tank of radius \(1 m\) is filled with water. The top surface of water is at \(15\,m\) from the bottom of the cylinder. There is a hole on the wall of cylinder at a height of \(5\,m\) from the bottom. A force of \(5 \times 10^{5} N\) is applied an the top surface of water using a piston. The speed of efflux from the hole will be. (given atmospheric pressure \(P_{A}=1.01 \times 10^{5}\,Pa\), density of water \(\rho_{ w }=1000\,kg / m ^{3}\) and gravitational acceleration \(g=10\,m / s ^{2}\) )

An alternating current is given by \(\mathrm{I}=\mathrm{I}_{\mathrm{A}} \sin \omega \mathrm{t}+\mathrm{I}_{\mathrm{B}} \cos \omega \mathrm{t}\). The r.m.s current will be ______.