A cylinder with fixed capacity of \(67.2\, lit\) contains helium gas at \(STP\). The amount of heat needed to raise the temperature of the gas by \(20\,^oC\) is ..... \(J\) [Given that \(R = 8.31\, J\, mol^{-1}\, K^{-1}\)]

A point charge of magnitude \(+ 1\,\mu C\) is fixed at \((0, 0, 0) \). An isolated uncharged spherical conductor, is fixed with its center at \((4, 0, 0).\) The potential and the induced electric field at the centre of the sphere is

A particle is moving in a circular path of radius \(a,\) with a constant velocity \(v\) as shown in the figure.The centre of circle is marked by \('C'\). The angular momentum from the origin \(O\) can be written as

A metal wire of length \(0.5 m\) and cross-sectional area \(10^{-4}\; m ^{2}\) has breaking stress \(5 \times 10^{8}\; Nm ^{-2}\). A block of \(10\; kg\) is attached at one end of the string and is rotating in a horizontal circle. The maximum linear velocity of block will be \(ms ^{-1}\).

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\) Assertion \(A\): Steel is used in the construction of buildings and bridges. Reason \(R:\) Steel is more elastic and its elastic limit is high. In the light of above statements, choose the most appropriate answer from the options given below

Two balls \(A\) and \(B\) are placed at the top of \(180 \,m\) tall tower. Ball \(A\) is released from the top at \(t =0 \,s\). Ball \(B\) is thrown vertically down with an initial velocity \(u\)' at \(t=2\, s\). After a certain time, both balls meet \(100 \,m\) above the ground. Find the value of \(u\) in ............... \(ms ^{-1}\). [use \(g =10 \,ms ^{-2}\) ]

A block is simply released from the top of an inclined plane as shown in the figure above. The maximum compression in the spring when the block hits the spring is _______.

A small bob A of mass m is attached to a massless rigid rod of length 1m pivoted at point P and kept at an angle of \( 60^{\circ} \) with vertical as shown in figure. At distance of 1m below point P, an identical bob B is kept at rest on a smooth horizontal surface that extends to a circular track of radius R as shown in figure. If bob B just manages to complete the circular path of radius R upto a point Q after being hit elastically by both A, then radius R is ___________m.

Figure represents the extension (\(\Delta l\)) of a wire of length \(1\) meter, suspended from the ceiling of the room at one end with a load \(W\) connected to the other end. If the cross-sectional area of the wire is \(10^{-5}\) m\(^2\) then the Young's modulus of the wire is __________ N/m\(^2\).

The total number of turns and cross-section area in solenoid is fixed. However, its length \(L\) is varied by adjusting the separation between windings. The inductance of solenoid will be proportional to

Charges \(Q _{1}\) and \(Q _{2}\) arc at points \(A\) and \(B\) of a right angle triangle \(OAB\) (see figure). The resultant electric field at point \(O\) is perpendicular to the hypotenuse, then \(Q _{1} / Q _{2}\) is proportional to

A stone of mass \(20\, {g}\) is projected from a rubber catapult of length \(0.1\, {m}\) and area of cross section \(10^{-6} \,{m}^{2}\) stretched by an amount \(0.04\, {m}\). The velocity of the projected stone is \(....\,m\,/s.\) (Young's modulus of rubber \(=0.5 \times 10^{9}\, {N} / {m}^{2}\) )