Four identical particles of equal masses $1\text{kg}$ are made to move along the circumference of a circle of radius $1\text{m}$ under the action of their own mutual gravitational attraction. The speed of each particle will be :

$[$Assume particles are at the vertices of a square, also $G$ is Gravitational Constant$]$

Equation for a wave on a string is given by $y=0.03\sin (15.70x-2t)$ where $y$ and $x$ are in metres and $t$ is in seconds. At $t=0$, find the value of displacement for a particle on string at $x=0.1\text{m}$.

Rain, driven by the wind, falls on a railway compartment with a velocity of $20m/s$, at an angle of ${30}^{\circ }$ to the vertical. The train moves, along the direction of wind flow, at a speed of $108km/hr$. Determine the apparent velocity of rain for a person sitting in the train.

The uniform electric field in a region surrounding the origin is along the $x-$axis. A small circle is drawn with the center at the origin cutting the axes at points $\text{A, B, C}$ and $\text{D}$ having coordinates $(a,0),(0,a),(-a,0)$ and $(0,-a)$ respectively, as shown in figure. Find the point at which the potential is minimum.

An iron rod of volume ${10}^{-4}{\text{m}}^3$ and relative permeability $1000$ is placed inside a long solenoid wound with $5\text{turns/cm}$. If a current of $0.5\text{A}$ is passed through the solenoid, find the magnetic moment of the rod.

A particle is released from rest at origin. It moves under the influence of a potential field, $U=x^2-3x$. Find the Kinetic energy of the particle at $x=2$.

When a man climbs up one-third of a ladder inclined at an angle $\theta$ to the ground, he observes an object on the ground at an angular depression of $\alpha$. If the angular depression of the same object when he climbs the ladder completely is $\beta ,$ then

The earth radiates in the infra-red region of the spectrum. The spectrum is correctly given by

Two particles are joined together by a string of total length $2L$ as shown in figure. Both the particles are performing uniform circular motion about point $\text{O}$ in a horizontal plane. If the speed of the outermost particle is $v_0$. Then the ratio of tension in string $\text{AB}$ and $\text{OA}$ will be:

A rope of mass M and length L is hanging from a ceiling of a room. Assume that the mass is uniformly distributed along the length of the rope, find the tension in it at point P(as shown in figure) which is $\frac{L}{3}$ units away from the top end of the rope?

The separation between a node and the next antinode in a vibrating air column is 25 cm. If the speed of sound in air is 340 $m{s}^{-1}$, find the frequency of vibration of the air column

A single cylinder oil engine has a stroke of 38 cm and crank rotates at 360 rpm. The reciprocating parts weigh 670 N and revolving parts are equivalent to 800 N at crank radius 'r'. A revolving balance weight is introduced at a radius of 15 cm to balance the whole of revolving and one half of reciprocating parts. The magnitude of balancing weight required and the residual unbalanced force on crank shaft are

Two blocks of masses $4\text{kg}$ and $2\text{kg}$ are connected by a light string and placed on rough horizontal plane. The $2\text{kg}$ block is pulled with a constant force $F$ as shown in the figure. The coefficient of friction between the blocks and the ground is $0.5$. What is the value of $F$ so that tension in the string is constant throughtout the motion of the blocks?

The current $i$ in a coil varies with time $t$ according to the graph shown below. Which of the following graphs, shows the induced emf $\left(E\right)$ in the coil with time?

Find the ratio of the force of interaction between the $4kg$ and $2kg$ blocks and $2kg$ and $3kg$ blocks. (Assume all surfaces to be smooth)

Question 24



Fig. shows a car sticking to an electromagnet. Name the forces acting on the car? Which one of them is larger?

Explain briefly the process of charging a parallel plate capacitor when it is connected across a d.c. battery.

A capacitor of capacitance ‘C’ is charged to ‘V’ volts by a battery. After some time the battery is disconnected and the distance between the plates is doubled. Now a slab of dielectric constant, 1 < k < 2, is introduced to fill the space between the plates. How will the following be affected?

(a) The electric field between the plates of the capacitor

(b) The energy stored in thecapacitor

Justify your answer by writing the necessary expressions

The density of a liquid of coefficient of cubical expansion $\gamma$ is $\rho$ at $0^{\circ }C$. When the liquid is heated to a temperature T, the change in density will be

Four particles, each of mass $M$, move along a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle is

(Assume that each particle is diameterically opposite to other in pair.)

A truck driver drives $20km$ down the road in 5 minutes. He then reverses and drives $12km$ back down the road in 3 minutes. What is his average velocity?

The focal lengths of the objective and of the eye-piece of a compound microscope are f_o and f_e respectively. If L is the tube length and D is the least distance of distinct vision, then its angular magnification, when the image is formed at infinity, is

Two slits are separated by $0.3\text{mm}$. A beam of $500\text{nm}$ light strikes the slits, producing an interference pattern. The number of maxima observed in the angular range $-{30}^{\circ }<\theta <{30}^{\circ }$.

How are the image formed when an object is moved from infinity to convex lens

Orbit of a planet around a star is

A coin lies at the bottom of a lake 2 m deep, at a horizontal distance 'x' from the viewer situated 1 m above the surface of a liquid as shown in the figure. Refractive index of water is $\mu =\sqrt{2}$. Find the value of x, if refracted ray enters the eye of the viewer at an angle of ${45}^{\circ }$ from the vertical.

If the potential energy between two molecules is given by $U=-\frac{A}{r^6}+\frac{B}{r^{12}},$ then at equilibrium, separation between molecules and the potential energy respectively are:

Two charges +q and -q are palced fixed on the corner of a massless rigid rod of length 2L. Calculate the potential energy of the dipole thus formed.

A conducting loop of area $5.0{\text{cm}}^2$ is placed in a magnetic field which varies sinusoidally with time as $B=B_0\sin \omega t$ where $B_0=0.20\text{T}$ and $\omega =300s^{-1}$. The normal to the coil makes an angle of ${60}^{\circ }$ with direction of the field. Find the maximum emf induced in the coil.

A simple pendulum of length L having a bob of mass m is deflected from its rest position by an angle $\theta$ and released (figure) The string hits a peg which is fixed at a distance x below the point of suspension and the bob starts going in a circle centred at the peg. If the pendulum is released with $\theta ={90}^{\circ }$ and x = $\frac{L}{2}$ find the maximum height reached by the bob above its lowest position before the string becomes slack.