Under isothermal condition, the pressure of a gas is given by \(P = aV ^{-3}\), where \(a\) is a constant and \(V\) is the volume of the gas. The bulk modulus at constant temperature is equal to \(..........\,P\)

In an experiment to determine the Young's modulus, steel wires of five different lengths \((1,2,3,4\) and \(5\,m )\) but of same cross section \(\left(2\,mm ^{2}\right)\) were taken and curves between extension and load were obtained. The slope (extension/load) of the curves were plotted with the wire length and the following graph is obtained. If the Young's modulus of given steel wires is \(x \times 10^{11}\,Nm ^{-2}\), then the value of \(x\) is

An elastic spring under tension of \(3 \mathrm{~N}\) has a lengtha. Its length is \(b\) under tension \(2 \mathrm{~N}\). For its length\((3 a-2 b)\), the value of tension will be_______. \(\mathrm{N}\).

The rms speed of oxygen molecule in a vessel at particular temperature is \(\left(1+\frac{5}{x}\right)^{\frac{1}{2}} v\), where \(v\) is the average speed of the molecule. The value of \(x\) will be:(Take \(\pi=\frac{22}{7}\) )

A part of a complete circuit is shown in the figure. At some instant, the value of current \(I\) is \(1\, A\) and it is decreasing at a rate of \(10^{2}\, A s ^{-1}\) The value of the potential difference \(V _{ P }- V _{ Q }\) (in volts) at that instant, is.

Two infinite identical charged sheets and a charged spherical body of charge density ' \(\rho\) ' are arranged as shown in figure. Then the correct relation between the electrical fields at \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and D points is :

The ranges and heights for two projectiles projected with the same initial velocity at angles \(42^{\circ}\) and \(48^{\circ}\) with the horizontal are \({R}_{1}, {R}_{2}\) and \({H}_{1}\), \({H}_{2}\) respectively. Choose the correct option:

The average translational kinetic energy of \({N}_{2}\) gas molecules at \(\ldots \ldots \ldots . .{ }^{\circ} {C}\) becomes equal to the \({K} . {E}\). of an electron accelerated from rest through a potential difference of \(0.1\) \(volt.\) \(\left(\right.\) Given \(\left.{k}_{{B}}=1.38 \times 10^{-23} \, {J} / {K}\right)\) (Fill the nearest integer).

Two waves are simultaneously passing through a string and their equations are : \({y}_{1}={A}_{1} \sin {k}({x}-v {t}), {y}_{2}={A}_{2} \sin {k}\left({x}-{vt}+{x}_{0}\right) .\) Given amplitudes \({A}_{1}=12\, {mm}\) and \({A}_{2}=5\, {mm}\) \({x}_{0}=3.5\, {cm}\) and wave number \({k}=6.28\, {cm}^{-1}\). The amplitude of resulting wave will be \(......\,{mm}\)

The percentage increase in the speed of transverse waves produced in a stretched string if the tension is increased by \(4\, \%\), will be ......... \(\%\)

Two particles \(A, B\) are moving on two concentric circles of radii \(R_1\) and \(R_2\) with equal angular speed \(\omega \). At \(t = 0,\) their positions and direction of motion are shown in the figure The relative velocity \(\overrightarrow {{V_A}}  - \overrightarrow {{V_B}} \) at \(t = \frac{\pi }{{2\omega }}\) is given by