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A tuning fork vibrating with a sonometer having $20cm$ wire produces $5$ beats per second. If the beat frequency does not change if the length of the wire is changed to $21cm$, then the frequency of the tuning fork $($in $Hz)$ must be

Two bodies of masses $M_1$ and $M_2$ are placed at a distance $R$ apart. Then at the position where the gravitational field due to them is zero, the gravitational potential is

Two long, thin, parallel conductors are kept very close to each other, without touching. One carries a current i, and the other has charge $\lambda$ per unit length. An electron moving parallel to the conductors is un-deflected. Let C = velocity of light. Then,

An object is kept at a distance of 10 cm from a concave mirror of focal length 5 cm, the magnitude of the image is

Find out the value of resistance $R$ in figure.

The equation of motion of a particle executing simple harmonic motion is $a+16{\pi }^{2}x=0$. In this equation, $a$ is the linear acceleration (in ${\text{m/s}}^2$) of the particle at a displacement $x$ (in meter). The time period in simple harmonic motion is

A thin spherical conducting shell of radius $R$ has a charge $q$. Another charge $Q$ is placed at the center of the shell. The electrostatic potential at a point $P$ at a distance $R/2$ from the center of the shell is

A block of mass $10\text{kg}$ is suspended from a string of length $4\text{m}$. When pulled by a force $F$ along the horizontal from its midpoint, upper half of the string makes an angle ${45}^{\circ }$ with the vertical. The value of $F$ is

A small block is placed on a turn table. The coefficient of static friction between the block and the table is ${\mu }_s=0.5$. If the turn table is rotating with constant angular acceleration $\alpha =3{\text{rad/s}}^2$, what will be the angular speed at which the block begins to slip on the table ?

Given $r=1\text{m},g=10{\text{m/s}}^2.$

One end of a light spring of force constant K is fixed to a block of mass M placed on a horizontal frictionless surface, the other end of the spring being fixed to a wall. The spring – block system is executing simple harmonic motion of amplitude A and frequency v. When the block is passing through the equilibrium position, an object of a mass m is gently placed on the block. As a result, the frequency of the system becomes v' and the amplitude becomes A' . The ratio v'/v will be,

A rigid wire loop of square shape having side of length $L$ and resistance $R$ is moving along the $x$-axis with a constant velocity $v_0$ in the plane of the paper. At $t=0$, the right edge of the loop enters a region of length $3L$ where there is a uniform magnetic field $B_0$ into the plane of the paper, as shown in the figure. For sufficient large $v_0$, the loop eventually crosses the region. Let $x$ be the location of the right edge of the loop. Let $v\left(x\right),I\left(x\right)$ and $F\left(x\right)$ represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of $x$. Counter-clockwise current is taken as positive.

Which of the following schematic plot(s) is (are) correct? (Ignore gravity)

Four particles of masses 1 kg, 2 kg, 3 kg and 4 kg are placed at the corners of a square of side 2 m in the x-y plane as shown in the figure. If the origin of the coordinate system is taken at the mass of 1 kg, the (x, y) co-ordinates of the centre of mass are

1. But the vessel kept going away.

A. He looked anxiously around.
B. There was nothing to see but the water and empty sky.
C. He could now barely see her funnel and masts when heaved up on a high wave.
D. He did not know for what.

A $40\text{cm}$ long wire having a mass $3.2\text{gm}$ and area of cross section $1{\text{mm}}^2$ is stretched between two supports $40.05\text{cm}$ apart. In its fundamental mode, it vibrates with a frequency $\frac{1000}{64}\text{Hz}$. Find the Young's modulus of the wire.

A conducting disc of radius $0.6\text{m}$ is in pure rolling motion. A magnetic field of $1\text{T}$ exists in the region along the axis of the disc. If the velocity of the center of the disc is $5\text{m/s}$, calculate the potential difference between the center and a point on its periphery.

Consider a coaxial cable which consists of an inner wire of radius $a$ surrounded by an outer shell of inner and outer radii $b$ and $c$ respectively. The inner wire carries an electric current $i$ and the outer shell carries an equal current in the opposite direction as shown in the figure. The magnetic induction at a distance $r$ from the axis, for $(b<r<c)$ is-

[Take permeability of the conductor as $\mu$]

A signal conditioning circuit suitable for a push-pull type capacitive transducer is given in figure. The output $V_0$ is

Under what conditions current passing through a resistance $R$ can be increased by short-circuiting the battery of emf $E_2$. The internal resistances of the two batteries are $r_1$ and $r_2$ respectively.

Figure (15-E10) shows an aluminium wire of length 60 cm joined to a steel wire of length 80 cm and stretched between two fixed supports. The tension produced is 40 N. The cross-selectional area of the steel wire is $1.0m{m}^2$ and that of the aluminium wire is $3.0m{m}^2$. What could be the minimum frequency of a tuning fork which can produce standing waves in the system with the joint as a node ? The density of aluminium is $2.6gc{m}^{-3}$ and that of steel is $7.8gc{m}^{-3}$

Figure shows two blocks $4kg$ and $2kg$ placed on a frictionless surface and connected with a spring. An external push gives a velocity of $10m{s}^{-1}$ to the heavier block in the direction of the lighter one. At that instant, the velocity gained by the centre of mass will be

Centre of mass of a system of three particles of masses $1\text{kg}$, $2\text{kg}$ and $3\text{kg}$ at the point $(1\text{m},2\text{m},3\text{m})$ and centre of mass of another group of two particles of masses $2\text{kg}$ and $3\text{kg}$ is at point $(-1\text{m},-2\text{m})$, where a $10\text{kg}$ particle should be placed, so that the centre of mass of the system of all those six particles shifts to centre of mass of the first system?

Twelve resistors each of resistance $r$ are connected together so that each lies along the edge of the cube as shown in the figure. The equivalent resistance between points $1$ and $4$ is

A symmetric star shaped conducting wire loop is carrying a steady state current $I$ as shown the figure. The distance between the diametrically opposite vertices of the star is $4a$. The magnitude of the magnetic field at the center of the loop is

A hollow vertical cylindrical structure rotates about its axis. Tom is standing on a horizontal platform against inner wall of cylindrical structure. At a certain angular speed of hollow cylinder, the horizontal platform below him is removed but he remains stationary against the inner wall. If the radius of the hollow cylinder is $5\text{m}$ and the coefficient of static friction between the cylindrical wall and Tom is $0.15$, find the minimum angular speed of the cylindrical structure at which the platform may be removed.

[Take $g=10{\text{m/s}}^2$]

A man is looking at a small object placed at his near point. Without altering the position of his eye or the object, he puts a simple microscope of magnifying power $5\text{X}$ before his eyes. The angular magnification achieved is

A two-fold increase in intensity of a wave implies an increase of (Given: ${\text{log}}_{10}2=0.3010$)

A particle starts from $x=0$ and moves along a straight line such that its velocity $v$ depends on position $x$ as $v=x+4$.

For the first four seconds, average speed of the particle (in $m/s$) is

A ray of light falls on a transparent sphere with centre at C as shown in figure. The ray emerges from the sphere parallel to line AB. The refractive index of the sphere is:

The product of the hole concentration and the conduction electron concentration turns out to be independent of the amount of any impurity doped. The concentration of conduction electrons in germanium is $6\times {10}^{19}$ per cubic metre. When some phosphorus impurity is doped into a germanium sample, the concentration of conduction electrons increases to $2\times {10}^{23}$ per cubic metre. Find the concentration of the holes in the doped germanium.

A simple pendulum of length $100\text{cm}$ is suspended from the ceiling of a cart which is sliding without friction on an inclined plane of inclination ${60}^{\circ }$. Find the time period of the pendulum.
Take $g=10{\text{m/s}}^2$.