Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $\alpha$. The spheres are now immersed in a dielectric liquid of density $800 \mathrm{kg} {\mathrm{m}}^{-3}$ and dielectric constant $21$. If the angle between the strings remains the same after the immersion, then

A train $\mathrm{S}1,$ moving with a uniform velocity of $108\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h},$ approaches another train $\mathrm{S}2$ standing on a platform. An observer $\mathrm{O}$ moves with a uniform velocity of $36\mathrm{}\mathrm{k}\mathrm{m}/\mathrm{h}$ towards $\mathrm{S}2,$ as shown in figure. Both the trains are blowing whistles of same frequency $120\mathrm{}\mathrm{H}\mathrm{z}.$ When $\mathrm{O}$ is $600\mathrm{}\mathrm{m}$ away from $\mathrm{S}2$ and distance between $\mathrm{S}1$ and $\mathrm{S}2$ is $800\mathrm{}\mathrm{m},$ the number of beats heard by $\mathrm{O}$ is____
[Speed of the sound $=330\mathrm{}\mathrm{m}/\mathrm{s}]$

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A uniform thin cylindrical disk of mass \(M\) and radius \(R\) is attached to two identical massless springs of spring constant \(k\) which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance \(d\) from its centre. The axle is massless and both the springs and the axle are in a horizontal plane. The unstretched length of each spring is \(L\). The disk is initially at its equilibrium position with its centre of mass \((C M)\) at a distance Lfrom the wall. The disk rolls without slipping with velocity \(\mathbf{v}_0=v_0 \hat{\mathbf{i}}\) The coefficient of friction is \(\mu\).
Question :
The maximum value of \(v_0\) for which the disk will roll without slipping is

An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure).Initially the gas is at temperature ${\mathrm{T}}_1$ , pressure ${\mathrm{P}}_1$ and volume ${\mathrm{V}}_1$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature ${\mathrm{T}}_2$ , pressure ${\mathrm{P}}_2$ and volume ${\mathrm{V}}_2$ . During this process the piston moves out by a distance $\mathrm{x}$ . Ignoring the friction between the piston the cylinder, the correct statement(s) is (are)

The figure shows the \(p-V\) plot of an ideal gas taken through a cycle \(A B C D A\). The part \(A B C\) is a semi-circle and \(C D A\) is half of an ellipse. Then,

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $v_n$. At a fixed temperature and pressure, which of the following is the correct option?

A block \((B)\) is attached to two unstretched \(S_1\) and \(S_2\) with spring constants \(k\) and \(4 k\), respectively (see figure I). The other ends are attached to identical supports \(M_1\) and \(M_2\) not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block \(B\) is displaced towards wall 1 by a small distance \(x\) (figure II) and released. The block returns and moves a maximum distance \(y\) towards wall 2 . Displacements \(x\) and \(y\) are measured with respect to the equilibrium position of the block \(B\). The ratio \(\frac{y}{-1}\) is

An ellipse intersects the hyperbola \(2 x^2-2 y^2=1\) orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

A compound \(\mathrm{M}_{\mathrm{p}} \mathrm{X}_{\mathrm{q}}\) has cubic close packing (ccp) arrangement of \(\mathrm{X}\). Its unit cell structure is shown below. The empirical formula of the compound is

Let $f\text{ : }\left[\frac{1}{2},1\right]⟶R$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{\text{'}}\left(x\right)<2f\left(x\right)$ and $f\left(\frac{1}{2}\right)=1$. Then the value of $\underset{1/2}{\overset{1}{\int }}f\left(x\right)\mathit{\text{dx}}$ lies in the interval:

Reaction of \(\mathrm{Br}_2\) with \(\mathrm{Na}_2 \mathrm{CO}_3\) in aqueous solution gives sodium bromide and sodium bromate with evolution of \(\mathrm{CO}_2\) gas. The number of sodium bromide molecules involved in the balanced chemical equation is

Isomers of hexane, based on their branching, can be divided into three distinct classes as shown in the figure.

The correct order of their boiling point is

In the circuit shown below, the switch \(\mathrm{S}\) is connected to position \(\mathrm{P}\) for a long time so that the charge on the capacitor becomes \(q_{1} \mu \mathrm{C}\). Then \(\mathrm{S}\) is switched to position \(\mathrm{Q}\). After a long time, the charge on the capacitor is \(q_{2} \mu \mathrm{C}\).

The magnitude of $q_1$ is _______.