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Figure shows three concentric thin spherical shells $A,B$ and $C$ of radii $R,2R$ and $3R$. The shell $B$ is earthed and $A$ and $C$ are given charges $q$ and $2q$ respectively. Find the charges appearing on the outer surface of $C$.

A small block of mass ‘m’ is rigidly attached at ‘P to a ring of mass ‘3m’ and radius ‘r’. The system is released from rest at $\theta ={90}^{\circ }$ and rolls without sliding. The angular acceleration of hoop about the center of the ring, just after the release is

Assume that a tunnel is dug across the earth (radius = R) passing through its centre. Find the time a particle takes to cover the length of the tunnel if it is projected into the tunnel with a speed of $\sqrt{gR}$

1. 15

A punching machine is used to punch a circular hole of diameter two units from a square sheet of aluminum of width 2 units, as shown below. The hole is punched such that the circular hole touches one corner P of the square sheet and the diameter of the hole originating at P is in line with a diagonal of the square.

The proportion of the sheet area that remains after punching is

The magnetic field through a single loop of wire having radius $12\text{cm}$ and resistance $8.5\Omega$ changes with time as shown in the figure. The magnetic field is perpendicular to the plane of the loop. Plot the induced current as a function of time.

The efficiency of ideal heat engine working between ${32}^{0}F$ and ${212}^{0}F$ is

The potential energy U between two molecules as a function of the distance X between them has been shown in the figure. The two molecules are

A boy and a block, both of same mass $m\text{kg}$, are initially at the same horizontal level and attached to a frictionless pulley through a light inextensible string that moves over the frictionless pulley as shown in figure. The boy starts moving upward with an acceleration of $5{\text{m/s}}^2$ relative to the rope. If the block has to travel a total distance of $10\text{m}$ before reaching at the level of pulley, the time taken by the block in doing so is equal to

The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed $\omega$ and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed $\frac{\omega }{2}$. The ring and disc are separated by frictionless ball bearings. The system is in the x - z place. The point P on the inner disc is at distance R from the origin O, where OP makes an angled of 30${}^{\circ }$ with the horizontal. Then with respect to the horizontal surface,

Two identical wires A and B, each of length '$l$', carry the same current $I$. Wire A is bent into a circle of radius $R$ and wire B is bent to form a square of side '$a$'. If $B_A$ and $B_B$ are the values of magnetic field at the centres of the circle and square respectively, then the ratio $\frac{B_A}{B_B}$ is

The tube is open at end $A$. Then, absolute pressure at end $B$ is - (Take atmospheric pressure to be ${10}^5\text{Pa}$)

Find the co-ordinates of the center of mass of a hollow cone having a height of $90\text{cm}$ as shown in the figure.

Which of the following gases can be most easily liquified? Given, the value of $a$ in van der Waals equation for $N{H}_3=4.17,C{O}_2=3.59,S{O}_2=6.71\&C{l}_2=6.49$

A cyclotron is used to accelerate

A piece of ice starting from rest slides down a rough ${45}^{\circ }$ incline in twice the time it takes to slide down a frictionless ${45}^{\circ }$ incline. What is the coefficient of friction between the ice and incline?

A body of mass $m$ thrown horizontally with velocity $v$, from the top of a tower of height $h$ touches the level ground at a distance of $250\text{m}$ from the foot of the tower. A body of mass $2m$ thrown horizontally with velocity $\frac{v}{2}$ from the top of a tower of height $4h$ will touch the level ground at a distance $x$ from the foot of tower. The value of $x$ is

A parallel plate capacitor is made of two circular plates separated by an distance of 5 mm and with a dielectric of dielectric constant 2.2 between them. When the electric field in the dielectric is $3\times {10}^{4}V/m$, the charge density of the positive plate will be close to:

Figure (9-E22) shows a small body of mass m placed over a larger mass M whose surface is horizontal near the smaller mass and gradually curves to become vertical. The smaller mass is pushed on the longer one at a speed v and the system is left to itself. Assume that all the surfaces are frictionless. (a) Find the speed of the larger block when the smaller block is sliding on the vertical part. (b) Find the speed of the smaller mass when it breaks off the larger mass at height h. (c) Find the maximum height (from the ground) that the smaller mass ascends (d) Show that the smaller mass will again land on the bigger one. Find the distance traversed by the bigger block during the time when the smaller block was in its fight under gravity.

Molecules of a gas behave like

If a combination of 3 convex lens is arranged in the shown manner the focal length of lens 1 is $+5cm$, lens II is $10cm$ and lens III is $20cm$. Calculate the total power of the combination of the lens.

1. 2.5

If $y=2si{n}^2\theta +tan\theta$, then $\frac{dy}{d\theta }$ will be

In a photoelectric experiment, electrons are ejected from metals X and Y by light of intensity I and frequency f. The potential difference V required to stop the electrons is measured for various frequencies. If Y has a greater work function than X; which one of the following graphs best illustrates the expected results?

Refer to the inverted slider crank mechanism shown in figure, at the instant the input crank (PS) is horizontal and perpendicular to fixed link and has an angular velocity of 1.25 rad/sec. The magnitude of the Coriolis acceleration is

The distance between the two charges +q and -q of a dipole is r. On the axial line at a distance d from the centre of dipole, the intensity is proportional to

A point $P$ is the contact point of wheel on the ground which rolls on ground without slipping. Find the value of displacement of the point $P$ when wheel completes half revolution. Radius of the wheel is $1\text{m}$.

Unpolarized light of intensity $I$ passes through an ideal Polarizer $A$. Another identical polarizer $B$ is placed behind $A$. The intensity of light beyond $B$ is found to be $\frac{I}{2}$. Now another identical polarizer $C$ is placed between $A$ and $B$. The intensity beyond $B$ is now found to be $\frac{I}{8}$. The angle between polarizer $A$ and $C$ is

A boat which has a speed of $5\text{km/h}$ in still water crosses a river of width $1\text{km}$ along the shortest possible path in $15\text{min}$. The magnitude of velocity of the river water in $\text{km/h}$ is

A thin square plate $ABCD$ of uniform thickness is shown in the figure. $I_1,I_2,I_3$ and $I_4$ are respectively the moments of inertia about axis $1,2,3$ and $4$ which are in the plane of the square plate. Then, the moment of inertia $I_0$ of the plate about an axis passing through its centre $\text{O}$ and perpendicular to the plane of the plate will be equal to : (Choose the correct option(s))

The figure shows a rectangular loop of wire immersed in a non-uniform and varying magnetic field $\overrightarrow{B}$ that is perpendicular to and directed into the page. The field's magnitude is given by $B=4t^2{x}^2$, with $B$ in $\text{tesla}$, $t$ in $\text{second}$, and $x$ in $\text{metre}$. The loop has width $W=3.0\text{m}$ and height $H=2.0\text{m}$. What is the magnitude of the induced emf $E$ around the loop at $t=0.10\text{s}$ ?

The velocity-time graph of a particle is given below. What is the acceleration of the particle?

Two masses m_{​​​​​​1} and m_{​​​​​​2} are connected by a spring of spring constant k and are placed on a smooth horizontal surface . Initially the spring is stretched through a distance d when the system is released from rest. Find the distance moved by the two masses when spring is compressed by a distance d .