A solid sphere is in pure rolling motion on an inclined surface having inclination \(\theta\).

An incompressible liquid is kept in a container having a weightless piston with a hole. A capillary tube of inner radius $0.1\mathrm{mm}$ is dipped vertically into the liquid through the airtight piston hole, as shown in the figure. The air in the container is isothermally compressed from its original volume $V_0$to$\frac{100}{101}{V}_0$with the movable piston.Considering air as an ideal gas, the height $\left(h\right)$ of the liquid column in the capillary above the liquid level in $\mathrm{cm}$is _____.
[Given: Surface tension of the liquid is $0.075\mathrm{N}{\mathrm{m}}^{-1}$, atmospheric pressure is ${10}^5\mathrm{N}{\mathrm{m}}^{-2}$, acceleration due to gravity $\left(g\right)$ is $10\mathrm{m}{\mathrm{s}}^{-2}$, density of the liquid is ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$and contact angle of capillary surface with the liquid is zero]

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is fixed vertically on ground as shown in the figure. The bead is released from near the top of the wire and it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is

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A point charge \(Q\) is moving in a circular orbit of radius \(R\) in the \(x-y\) plane with an angular velocity \(\omega .\) This can be considered as equivalent to a loop carrying a steady current \(\frac{Q \omega}{2 \pi}\). A uniform magnetic field along the positive \(z\)-axis is now switched on, which increases at a constant rate from 0 to \(B\) in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant \(\gamma\).

Question:

The magnitude of the induced electric field in the orbit at any instant of time during the time interval of the magnetic field change is

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Scientists are working hard to develop nuclear fusion reactor. Nuclei of heavy hydrogen, \({ }_1^2 \mathrm{H}\) known as deuteron and denoted by \(D\) can be thought of as a candidate for fusion reactor. The \(D\) - \(D\) reaction is \({ }_1^2 \mathrm{H}+{ }_1^2 \mathrm{H} \rightarrow{ }_2^3 \mathrm{He}+n+\) energy. In the core of fusion reactor. A gas of heavy hydrogen is fully ionized into deuteron nuclei and electrons. This collection of \({ }_1^4 \mathrm{H}\) nuclei and electrons is known as plasma. The nuclei move randomly in the reactor core and occasionally come close enough for nuclear fusion to take place. Usually, the temperatures in the reactor core are too high and no material wall can be used to confine the plasma. Special techniques are used which confine the plasma for a time \(t_0\) before the particles fly away from the core. If n is the density (number/volume) of deutrons, the product t \(_0\) is called Lawson number. In one of the criteria, \(a\) reactor is termed successful if Lawson number is greater than \(5 \times 10^{14} \mathrm{scm}^{-3}\).
It may be helpful to use the following : Boltzmann constant \(k=8.6 \times 10^{-5} \mathrm{eV} / \mathrm{K}\); \(\frac{e^2}{4 \pi \varepsilon_0}=1.44 \times 10^{-9} \mathrm{eVm}\)
Question:
In the core of nuclear fusion reactor, the gas becomes plasma because of

A lamina is made by removing a small disc of diameter \(2 R\) from a bigger disc of uniform mass density and radius \(2 R\), as shown in the figure. The moment of inertia of this lamina about axes passing though \(O\) and \(P\) is \(I_{O}\) and \(I_{P}\) respectively. Both these axes are perpendicular to the plane of the lamina. The ratio \(I_{P} / I_{O}\) to the nearest integer is

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In the figure a container is shown to have a movable (without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of a thermally conducting material that allows slow transfer of heat. The lower compartment of the container is filled with \(2\) moles of an ideal monatomic gas at \(700 K\) and the upper compartment is filled with \(2\) moles of an ideal diatomic gas at \(400 K\). The heat capacities per mole of an ideal monatomic gas are \(C_{V}=\frac{3}{2} R, C_{P}=\frac{5}{2} R\), and those for an ideal diatomic gas are \(C_{V}=\frac{5}{2} R, C_{P}=\frac{7}{2} R\).


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Consider the partition to be rigidly fixed so that it does not move. When equilibrium is achieved, the final temperature of the gases will be

A \(2 \mu \mathrm{F}\) capacitor is charged as shown in the figure. The percentage of its stored energy dissipated after the switch \(S\) is turned to position 2 , is

The correct statement(s) about surface properties is(are)

Consider $4$ boxes, where each box contains $3$ red balls and $2$ blue balls. Assume that all $20$ balls are distinct. In how many different ways can $10$ balls be chosen from these $4$ boxes so that from each box at least one red ball and one blue ball are chosen?

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Electrical resistance of certain materials, known as superconductors, changes abruptly from a non-zero value to zero as their temperature is lowered below a critical temperature \(T_C(0)\). An interesting property of superconductors is that their critical temperature becomes smaller than \(T_C(0)\) if they are placed in a magnetic field i.e., the critical temperature \(T_C(B)\) is a function of the magnetic field strength \(B\). The dependence of \(T_C(B)\) on \(B\) is shown in the figure.

Question :
A superconductor has \(T_C(0)=100 \mathrm{~K}\). When a magnetic field of \(7.5\) Tesla is applied, its \(T_C\) decreases to \(75 \mathrm{~K}\). For this material one can definitely say that when (Note : \(\mathrm{T}\) = Tesla)

The specific heat capacity of a substance is temperature dependent and is given by the formula \(C=k T\), where \(k\) is a constant of suitable dimensions in SI units, and \(T\) is the absolute temperature. If the heat required to raise the temperature of \(1 \mathrm{~kg}\) of the substance from \(-73^{\circ} \mathrm{C}\) to \(27^{\circ} \mathrm{C}\) is \(n k\), the value of \(n\) is _____
[Given: \(0 \mathrm{~K}=-273^{\circ} \mathrm{C}\).]

Three plane mirrors form an equilateral triangle with each side of length $L$. There is a small hole at a distance $l>0$ from one of the corners as shown in the figure. A ray of light is passed through the hole at an angle $\theta$ and can only come out through the same hole. The cross section of the mirror configuration and the ray of light lie on the same plane.

Which of the following statement(s) is(are) correct?