Consider a uniform rod of mass \(\mathrm{M}=4 \mathrm{m}\) and length \(\ell\) pivoted about its centre. A mass \(m\) moving with velocity \(v\) making angle \(\theta=\frac{\pi}{4}\) to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is

A prism of angle \(75^{\circ}\) and refractive index \(\sqrt{3}\) is coated with thin film of refractive index 1.5 only at the back exit surface. To have total internal reflection at the back exit surface the incident angle must be ___________ .
\(\left(\sin 15^{\circ}=0.25\right.\) and \(\left.\sin 25^{\circ}=0.43\right)\)

For a free body diagram shown in the figure, the four forces are applied in the ' \(x\) ' and ' \(y\) ' directions. What additional force must be applied and at what angle with positive \(x\)-axis so that the net acceleration of body is zero?

The mean free path of molecules of a certain gas at \(STP\) is \(1500\,d\), where \(d\) is the diameter of the gas molecules. While maintaining the standard pressure, the mean free path of the molecules at \(373\,K\) is approximately \(..........\,d\)

A uniform disc of radius \(R\) and mass \(M\) is free to rotate only about its axis. A string is wrapped over its rim and a body of mass \(m\) is tied to the free end of the string as shown in the figure. The body is released from rest. Then the acceleration of the body is

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 sample of a liquid is kept at 1 atm. It is compressed to 5 atm which leads to change of volume of \(0.8 \mathrm{~cm}^3\). If the bulk modulus of the liquid is 2 GPa , the initial volume of the liquid was _______ litre. \(\left(\right.\) Take \(\left.1 \mathrm{~atm}=10^5 \mathrm{~Pa}\right)\)

Assuming the validity of Bohr's atomic model for hydrogen like ions the radius of \(\mathrm{Li}^{++}\)ion in its ground state is given by \(\frac{1}{X} a_0\), where \(X=\) _______. (Where \(\mathrm{a}_0\) is the first Bohr's radius.)

A simple pendulum made of a bob of mass \(m\) and a metallic wire of negligible mass has time period \(2s\) at  \(T = 0\,^oC\) . If the temeprature of the wire is increased and the corresponding change in its time peirod is plotted against its temperature, the resulting graph is a line of slope \(S\). If the coefficient of linear expansion of metal is \(\alpha \) then the value of  \(S\) is

\(A\). The drift velocity of electrons decreases with the increase in the temperature of conductor. \(B\). The drift velocity is inversely proportional to the area of cross-section of given conductor. \(C\). The drift velocity does not depend on the applied potential difference to the conductor. \(D\). The drift velocity of electron is inversely proportional to the length of the conductor. \(E\). The drift velocity increases with the increase in the temperature of conductor. Choose the correct answer from the options given below:

An alternating current is represented by the equation, \(i=100 \sqrt{2} \sin (100 \pi t)\) ampere. The RMS value of current and the frequency of the given alternating current are ________.