Q. A football of radius R is kept on a hole of radius $r(r<R)$ made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle $\theta$ from the horizontal as shown in the figure below. The maximum value of $\theta$ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

1. $\sin \theta =\frac{r}{R}$
2. $\tan \theta =\frac{r}{R}$
3. $\sin \theta =\frac{r}{2R}$
4. $\cos \theta =\frac{r}{2R}$

Q. A light disc made of aluminium (a nonmagnetic material) is kept horizontally and is free to rotate about its axis as shown in the figure. A strong magnet is held vertically at a point above the disc away from its axis. On revolving the magnet about the axis of the disc, the disc will (figure is schematic and not drawn to scale)

1. rotate in the direction opposite to the direction of magnet’s motion
2. rotate in the same direction as the direction of magnet’s motion
3. not rotate and its temperature will remain unchanged
4. not rotate but its temperature will slowly rise

Q. A small roller of diameter 20 cm has an axle of diameter 10 cm (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved 50 cm, the position of the scale will look like (figures are schematic and not drawn to scale)

Q. A circular coil of radius R and N turns has negligible resistance. As shown in the schematic figure, its two ends are connected to two wires and it is hanging by those wires with its plane being vertical. The wires are connected to a capacitor with charge Q through a switch. The coil is in a horizontal uniform magnetic field $B_0$ parallel to the plane of the coil. When the switch is closed, the capacitor gets discharged through the coil in a very short time. By the time the capacitor is discharged fully, magnitude of the angular momentum gained by the coil will be (assume that the discharge time is so short that the coil has hardly rotated during this time)

1. $\frac{\pi }{2}NQ{B}_0{R}^2$
2. $\pi NQ{B}_0{R}^2$
3. $2\pi NQ{B}_0{R}^2$
4. $4\pi NQ{B}_0{R}^2$

Q. A parallel beam of light strikes a piece of transparent glass having cross section as shown in the figure below. Correct shape of the emergent wavefront will be (figures are schematic and not drawn to scale)

Q. A particle of mass m moves in circular orbits with potential energy $V\left(r\right)=Fr$, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle’s orbit is denoted by R and its speed and energy are denoted by v and E, respectively, then for the nth orbit (here h is the Planck’s constant)

1. $R\propto n^{1/3}$ and $v\propto n^{2/3}$
2. $R\propto n^{2/3}$ and $v\propto n^{1/3}$
3. $E=\frac{3}{2}{\left(\frac{n^2{h}^2{F}^2}{4{\pi }^{2}m}\right)}^{1/3}$
4. $E=2{\left(\frac{n^2{h}^2{F}^2}{4{\pi }^{2}m}\right)}^{1/3}$

Q. The filament of a light bulb has surface area $64m{m}^2$. The filament can be considered as a black body at temperature $2500K$ emitting radiation like a point source when viewed from far. At night the light bulb is observed from a distance of 100 m. Assume the pupil of the eyes of the observer to be circular with radius 3 mm. Then
(Take Stefan-Boltzmann constant $=5.67\times {10}^{-8}W{m}^{-2}{K}^{-4}$, Wien' displacement constant= $2.90\times {10}^{-3}m-K$, Planck's constant $=6.63\times {10}^{-34}Js$ speed of light in vaccum $=3.00\times {10}^{8}m{s}^{-1}$)

1. power radiated by the filament is in the range 642 W to 645 W
2. radiated power entering into one eye of the observer is in the range $3.15\times {10}^{-8}W$ to $3.25\times {10}^{-8}W$
3. the wavelength corresponding to the maximum intensity of light is 1160 nm
4. taking the average wavelength of emitted radiation to be 1740 nm, the total number of photons entering per second into one eye of the observer is in the range $2.75\times {10}^{11}$ to $2.85\times {10}^{11}$

Q. Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows: [position] = $\left[position\right]=\left[X^{\alpha }\right];\left[speed\right]=\left[X^{\beta }\right];\left[acceleration\right]=\left[X^p\right];\left[linermomentum\right]=\left[X^q\right];\left[force\right]=\left[X^r\right]$. Then

1. $\alpha +p=2\beta$
2. $p+q-r=\beta$
3. $p-q+r=\alpha$
4. $p+q+r=\beta$

Q. A uniform electric field, $\overrightarrow{E}=-400\sqrt{3}yN{C}^{-1}$ is applied in a region. A charged particle of mass m carrying positive charge q is projected in this region with an initial speed of $2\sqrt{10}\times {10}^{6}m{s}^{-1}$. This particle is aimed to hit a target T, which is 5 m away from its entry point into the field as shown schematically in the figure.
Take $\frac{q}{m}={10}^{10}Ck{g}^{-1}$. Then

1. the particle will hit T if projected at an angle ${45}^0$ from the horizontal
2. the particle will hit T if projected either at an angle ${30}^o$ or ${60}^o$ from the horizontal
3. time taken by the particle to hit T could be $\sqrt{\frac{5}{6}}\mu s$ us as well as $\sqrt{\frac{5}{6}}\mu s$
4. time taken by the particle to hit T is $\sqrt{\frac{5}{3}}\mu s$

Q. Shown in the figure is a semicircular metallic strip that has thickness t and resistivity $\rho$ . Its inner radius is $R_1$ and outer radius is $R_2$. If a voltage $V_0$ is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage $\Delta V$ develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale)

1. $I=\frac{v_{0}t}{\pi \rho }ln\left(\frac{R_2}{R_1}\right)$
2. the outer surface is at a higher voltage than the inner surface
3. the outer surface is at a lower voltage than the inner surface
4. $\Delta V\propto I$

Q. As shown schematically in the figure, two vessels contain water solutions (at temperature 𝑇) of potassium permanganate $\left(KMn{O}_4\right)$ of different concentrations $n_1$ and $n_2$ ($n_1>n_2$) molecules per unit volume with $\Delta n=(n_1-n_2)<<n_1$. When they are connected by a tube of small length 𝑙 and cross-sectional area S, $KMn{O}_4$ starts to diffuse from the left to the right vessel through the tube. Consider the collection of molecules to behave as dilute ideal gases and the difference in their partial pressure in the two vessels causing the diffusion. The speed 𝑣 of the molecules is limited by the viscous force $-\beta v$ on each molecule, where $\beta$ is a constant. Neglecting all terms of the order $(\Delta \Delta n{)}^2$, which of the following is/are correct? ($k_B$ is the Boltzmann constant)

1. the force causing the molecules to move across the tube is $\Delta n{k}_{B}TS$
2. force balance implies $n_1\beta vl=\Delta n{k}_{B}T$
3. total number of molecules going across the tube per sec is $\left(\frac{\Delta n}{l}\right)\left(\frac{k_{B}T}{\beta }\right)S$
4. rate of molecules getting transferred through the tube does not change with time

Q. Put a uniform meter scale horizontally on your extended index fingers with the left one at 0.00 cm and the right one at 90.00 cm. When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance $x_R$ from the center (50.00 cm) of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are 0.40 and 0.32, respectively, the value of $x_R$ (in cm) is ______.

Q. When water is filled carefully in a glass, one can fill it to a height h above the rim of the glass due to the surface tension of water. To calculate h just before water starts flowing, model the shape of the water above the rim as a disc of thickness h having semicircular edges, as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are ${10}^{3}kg{m}^{-3},0.07N{m}^{-1}$ and $10m{s}^{-2}$, respectively, the value of h (in mm) is _________.