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Find the moment of inertia of two solid spherical balls of mass $m$, radius $r$ attached at the two ends of a thin straight rod of mass $m$ and length $4r$, if this system is free to rotate about an axis passing through an end of the rod and perpendicular to its length.

A cyclic process is shown in P - T diagram. Which of the curves show same process on P-V graph.

In the pulley arrangement shown below, the pulley $P_2$ is movable. Assuming the coefficient of friction between m and surface to be $\mu$, the maximum value of M for which m is at rest is

Find total heat developed per second in the circuit.

Two sound waves move in the same direction. If the average power transmitted across a cross-section by them are equal while their wavelengths are in the ratio of 1:2. Their pressure amplitudes would be in the ratio of ___.

Potential energy U varies with distance x as $U=\frac{A{x}^{1/2}}{X^2+B}$ where A and B are constant. The dimensional formula for $A\times B$ is

In below figure two blocks are placed on fixed inclined plane and friction coefficient between the blocks and inclined plane are ${\mu }_1$ and ${\mu }_2$ respectively as shown in figure. Value of ${\mu }_1$ and ${\mu }_2$ are given in column 1 and value of $a_1$ and $a_2$ are given in column 2 and column 3 respectively.

$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ \left(i\right)& {\mu }_1=0.8& \left(i\right)& a_1=2m/s^2& \left(P\right)& a_2=2m/s^2\\ & {\mu }_2=0.8& & & & \\ \left(ii\right)& {\mu }_1=0.5& \left(ii\right)& a_1=0.8m/s^2& \left(Q\right)& a_2=0.8m/s^2\\ & {\mu }_2=0.5& & & & \\ \left(iii\right)& {\mu }_1=0.5& \left(ii\right)& a_1=4.4m/s^2& \left(R\right)& a_2=4.4m/s^2\\ & {\mu }_2=0.2& & & & \\ \left(iv\right)& {\mu }_1=0.5& \left(iv\right)& a_1=0& \left(S\right)& a_2=0\\ & {\mu }_2=0.8& & & & \end{array}$

Which of the following combination is correct?

The variation of magnetic susceptibility $\left(\chi \right)$ with absolute temperature T

A body is falling under gravity from rest. When it loses gravitational potential energy by U, its speed is v. The mass of the body shall be

A spherical conductor A contains two spherical cavities. The total charge on the conductor itself is zero. However, there is a point charge $q_b$ at the centre of one cavity and $q_c$ at the centre of the other. A considerable distance r away from the centre of the spherical conductor, there is another charge $q_b$. Force acting on $q_b,q_c$ and $q_d$ are $F_1,F_2$ and $F_3$ respectively. [Assume all charges are positive]

As the switch $S$ is closed in the circuit as shown in figure, current passed through it is

Two rods $\text{A}$ and $\text{B}$ of different materials are welded together as shown in figure. Their thermal conductivities are $K_1$and $K_2.$ The thermal conductivity of the composite rod will be

A block reaches the foot of a smooth inclined plane by slipping down with speed $V$. Now, if a disc of same mass as the block is released from same height, then its speed when it reaches the foot of the same plane will be

Microwaves from a transmitter are directed normally towards a plane reflector. A detector moves along the normal to the reflector. Between positions of $15$ successive maxima the detector travels a distance of $0.14m$. The frequency of transmitter is $(c=3\times {10}^{8}m/s)$:

A force of $(3\hat{i}-1.5\hat{j})$ acts on 5 kg body. The body is at a position of $(2\hat{i}-3\hat{j})m$ and is travelling at $4m{s}^{-1}$.
The force acts on the body until it is at the position $(\hat{i}+5\hat{j})$ m. Assuming no other force does work on the body, the final speed of the body is

One mole of an ideal diatomic gas undergoes a transition from A to B along a path AB as shown in the figure below. The change in internal energy of the gas during the transition is :

A sphere of mass M and radius R is released from the top of an inclined plane of inclination $\theta$. The minimum coefficient of friction between the plane and the sphere so that it rolls down the plane without sliding is given by

A conductor MN of length 30 cm rotates about the end M at 1000 rpm in a plane making an angle $\theta$ with the direction of the magnetic field perpendicular to the plane of the paper and pointing upwards. The strength of the field is $0.5Wb/m^2$. The induced emf (in volts) in the rod is

Two point charges $q_1=2\mu C$ and $q_2=1\mu C$ are placed at distances b = 1 cm and a = 2 cm from the origin of the y and x axes as shown in figure. The electric field vector at point P(a, b) will subtend an angle $\theta$ with the x-axis given by $tan\theta =K$. Find value of K.___

A battery of 15V and of negligible internal resistance is connected to a rheostat XZ of 1 $k\Omega$. The resistance of YZ part is $500\Omega$. The reading of the ammeter will be

A magnetic needle lying parallel to a magnetic field requires $W$ unit of work to turn it through ${60}^{\circ }$. The torque needed to maintain the needle in this position is

A body oscillates harmonically with amplitude 0.05 m. At a certain instant of time its displacement is 0.01 m and acceleration is 1.0 $m{s}^{-2}$ . What is the period of oscillation ?

Masses $8\text{kg}$, $2\text{kg}$ , $4\text{kg}$ & $2\text{kg}$ are placed at the corners $\text{A, B, C, D}$ respectively of a square $\text{ABCD}$ of diagonal $80\text{cm}$ (as shown in the figure below). The distance of the centre of mass from $A$ will be

A $10\text{kg}$ block placed on a rough horizontal floor is being pulled by a constant force $F=100\text{N}$ acting at angle ${37}^{\circ }$. The coefficient of kinetic friction between the block and the floor is $0.4$. Find the total work done due to force $F$ acting on the block over the displacement of $5\text{m}$.

A square plate of edge $a/2$ is cut out from a uniform square plate of edge $a$ as shown in the figure. The mass of the remaining portion is $M$ . The moment of inertia of the shaded portion about an axis passing through $O$ (centre of the square of side $a$ ) and perpendicular to plane of the plate is

Two spheres $A$ and $B$ having radii $3\text{cm}$ and $5\text{cm}$ respectively are coated with carbon black on their outer surface. The wavelengths of maximum intensity of emission of radiation are $300\text{nm}$ and $500\text{nm}$ respectively. The respective powers radiated by them are in the ratio of

A uniform current carrying ring of mass $m$ and radius $R$ is connected by a massless string as shown in the figure. A uniform magnetic field $B_0$ exists in the region to keep the ring in horizontal position, then the current in the ring is ($l=$ length of string)

A sinusoidal voltage of peak value $200\text{V}$ is connected to a diode and resistor $R$ in the circuit shown so that half wave rectification occurs. If the forward resistance of the diode is negligible compared to $R$, the rms value of voltage across $R$ in volts is approximately

A force F is applied to a system of light pulleys to pull body A. If F is 10 kN and block of mass A has a mass of $5\times {10}^3$ kg. What is the speed of A after 1 second starting form rest? Assume no friction.

A particle moves in a circle of radius $3.5\text{cm}$ at a speed given by $v=8t$, where $v$ is in $\text{cm/s}$ and $t$ in seconds. Find the tangential acceleration at $t=2\text{s}$. (in ${\text{cm/s}}^2)$

The initial velocity of the particle is $10\text{m/sec}$ and its retardation is $2{\text{m/sec}}^2$. The distance moved by the particle in $5^{\text{th}}$ second of its motion is

Find the acceleration of block $A$ and $B$? Take $g=10m/s^2$

The pressure of the gas in a constant volume gas thermometer is 70 kPa at the ice point. Find the pressure at the steam point.

A bob of mass m is suspended from the ceiling of a train moving with an acceleration a as shown in figure. Find the angle $\theta$ in equilibrium position.

Given in the figure are two blocks A and B of weight $20N$ and $100N$ respectively. The blocks are being pressed against a wall by a force $F$ as shown. If the coefficient of friction between the blocks is $0.1$ and between block B and the wall is $0.15$, the frictional force applied by the wall on block B is

The temperature reading of a bath on Celsius and Fahrenheit thermometers are in the ratio $2:5$. The temperature of the bath is

Two point sound sources of same frequency 'f' amplitude 2A and 4A are situated at $x=0$ and $x=20\lambda$ respectivley $(\lambda$ = wavelength of wave) on the x - axis. These sources are vibrating in same phase. Four observers A,B,C and D are situated at $x_A=4\lambda ,x_B=8\lambda ,x_C=12\lambda ,x_D=16\lambda$ respectively. The value of intensity observed by another observer O situated exactly at midpoint of location of two sources is $I_0$. $(\text{Given}{I}_0=\frac{36A^{2}K}{100})$ List - I gives the above four observers.
$\begin{array}{cc}\text{List - I}& \text{List - II}\\ \text{(I) Observer A}& \left(P\right)\frac{9}{16}\\ \text{(II) Observer B}& \left(Q\right)\frac{49}{144}\\ \text{(III) Observer C}& \left(R\right)\frac{4}{9}\\ \text{(IV) Observer D}& \left(S\right)\frac{5}{16}\\ & \left(T\right)\frac{81}{64}\\ & \left(U\right)\frac{10}{36}\end{array}$

In separate experiment location of sources are interchanged. The values of intensity observed by the observers now in units of $K{A}^2$ are given in List-II. Match respective intensities observed by the observers.

Find the value of $M$ in terms of m if the system is in equilibrium.

Calculate the energy released when $1000$ small water drops each of same radius ${10}^{-7}\text{m}$ coalesce to form one large drop. The surface tension of water is $5\times {10}^{-2}\text{N/m}$.

Repeat the previous exercise if the angle between each pair of springs is ${120}^{\circ }$ initially.

A rocket is launched with gas ejection speed of $400\text{m/s}$ and acceleration of $25{\text{m/s}}^2$. If mass of the rocket is $4000\text{kg}$, what will be the rate of consumption of the fuel?
(Take $g=10{\text{m/s}}^2$)

One of the square faces of a metal slab of side $50\text{cm}$ and thickness $20\text{cm}$ is rigidly fixed on a horizontal surface. If a tangential force of $30\text{N}$ is applied to the top face, then the displacement (in $\text{m}$) of the top face is :
[The shear modulus of the material is $4\times {10}^{10}{\text{N/m}}^2$]

A body of mass $2kg$ is kept on a rough horizontal surface. If ${\mu }_s=0.5$, find the contact force between the body and the surface. Take $g=10m/s^2$.

The temperature of a gas at pressure $P$ and volume $V$ is ${27}^{\circ }\text{C}$. Keeping its volume constant, if its temperature is raised to ${927}^{\circ }\text{C}$, then its pressure will be -