charge \(Q\) is uniformly distributed over a long rod \(AB\) of length \(L\) as shown in the figure. The electric potential at the point \(O\) lying at distance \(L\) from the end \(A\) is

Match List - \(I\) with List - \(II.\)

List - \(I\)	List - \(II\)
\((a)\) Source of microwave frequency	\((i)\) Radioactive decay on nucleus
\((b)\) Source of infrared frequency	\((ii)\) Magnetron
\((c)\) Source of Gamma Rays	\((iii)\) Inner shell electrons
\((d)\) Source of \(X-\)rays	\((iv)\) Vibration of atoms and molecules
	\((v)\) \(LASER\)
	\((vi)\) \(RC\) circuit

Choose the correct answer from the options given below :

A cylinder is rolling down on an inclined plane of inclination \(60^{\circ}\). It's acceleration during rolling down will be \(\frac{\mathrm{x}}{\sqrt{3}} \mathrm{~m} / \mathrm{s}^2\), where \(\mathrm{x}=\)___________. (use \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) ).

A solid body of constant heat capacity \(1\ J/^o C\) is being heated by keeping it in contact with reservoirs in two ways : \((i)\) Sequentially keeping in contact with \(2\) reservoirs such that each reservoir supplies same amount of heat. \((ii)\) Sequentially keeping in contact with \(8\) reservoirs such that each reservoir supplies same amount of heat. In both the cases body is brought from initial temperature \(100^o C\) to final temperature \(200^o C\). Entropy change of the body in the two cases respectively is :

A tub is filled with water and a wooden cube \(10\) cm \(\times\) \(10\) cm \(\times\) \(10\) cm is placed in the water. The wooden cube is found to float on the water with a part of it submerged in water. When a metal coin is placed on the wooden cube, the submerged part is increased by \(3.87\) cm. The mass of the metal coin is _______ gram.
(Take water density as \(1\) g/cm\(^3\) and density of wood as \(0.4\) g/cm\(^3\))

A square aluminium (shear modulus is \(25 \times 10^{9}\,Nm ^{-2}\) ) slab of side \(60\,cm\) and thickness \(15\,cm\) is subjected to a shearing force (on its narrow face) of \(18.0 \times 10^{4}\,N\). The lower edge is riveted to the floor. The displacement of the upper edge is \(.......\mu\,m\).

A solid cylinder is released from rest from the top of an inclined plane of inclination \(30^{\circ}\) and length \(60\,cm\). If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is \(...........ms ^{-1}\). (Given \(g =10\,ms ^{-2}\))

A solid sphere, of radius \(R\) acquires a terminal velocity \(\nu_1 \) when falling (due to gravity) through a viscous fluid having a coefficient of viscosity \(\eta \). The sphere is broken into \(27\) identical solid spheres. If each of these spheres acquires a terminal velocity, \(\nu_2\), when falling through the same fluid, the ratio \((\nu_1/\nu_2)\) equals

An object ' \(o\) ' is placed at a distance of \(100\,cm\) in front of a concave mirror of radius of curvature \(200\,cm\) as shown in the figure. The object starts moving towards the mirror at a speed \(2\,cm / s\). The position of the image from the mirror after \(10\,s\) will be at ...... \(cm\).

Given below are two statements Statement \(I\) : When the frequency of an \(a.c.\) source in a series \(LCR\) circuit increases, the current in the circuit first increases, attains a maximum value and then decreases. Statement \(II:\) In a series \(LCR\) circuit, the value of power factor at resonance is one. In the light of given statements, choose the most appropriate answer from the options given below

A student is performing the experiment of resonance column. The diameter of the column tube is \(6 \,cm\). The frequency of the tuning fork is \(504\, Hz\). Speed of the sound at the given temperature is \(336\, m / s\). The zero of the meter scale coincides with the top end of the resonance column tube. The reading of the water level in the column when the first resonance occurs is..........\(cm\)

Hydrogen ion and singly ionized helium atom are accelerated, from rest, through the same potential difference. The ratio of final speeds of hydrogen and helium ions is close to......