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Consider a wheel purely rolling on a rough horizontal surface with constant velocity v. Radius of the wheel is R and C is the center of wheel. M is top-most point, P is bottom-most point and N is in level with C at any time. Match the columns for this instant of time.

Two particles of mass $1\text{kg}$ and $3\text{kg}$ move towards each other under their mutual force of attraction. No other forces act on them. When the velocity of approach of the two particles is $2\text{m/s}$, their centre of mass has a velocity of $0.5\text{m/s}$. When the velocity of approach becomes $3\text{m/s}$, the velocity of the centre of mass of the system of particles is

In a PN-junction diode [MP PET 1993]

An infinite number of bricks are placed one over the other as shown in the figure. Each succeeding brick having half the length and breadth of its preceding brick and the mass of each succeeding brick being ${\frac{1}{4}}^{\text{th}}$ of the preceding one. Take $O$ as the origin. The $x-$coordinate of centre of mass of the system of bricks is:

Two identical parallel plates are given charges as shown in figure. If plate area is A & plate seperation is d then heat liberated after closing switch s is.

A helicopter lifts a $72\text{kg}$ astronaut $15\text{m}$ vertically from the ocean by means of a cable. The acceleration of the astronaut is $\frac{g}{10}$. How much work is done on the astronaut by the force from the helicopter ?

Calculate the area of the parallelogram when adjacent sides are given by the vectors $\overrightarrow{A}$ = $\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{B}$ = $2\hat{i}-3\hat{j}+\hat{k}$

The efficiency of the Otto cycle is given by which of these expressions?Where r = compression ratio.

A concave meniscus is observed for a liquid kept in a capillary tube, then angle of contact can be

A lead bullet penetrates into a solid object and melts. Assuming that 50% of its kinetic energy was used to heat it, calculate the initial speed of the bullet. The initial temperature of the bullet is ${27}^{\circ }C$ and its melting point is ${327}^{\circ }C$. Latent heat of fusion of lead = $2.5\times {10}^{4}Jk{g}^{-1}$ and specific heat capacity of lead = $125Jk{g}^1{K}^1$.

A block $M$ shown in the figure oscillates in a simple harmonic motion with amplitude $A$. The amplitude of the point $P$ is

A sample of an ideal gas has diatomic molecules at a temperature at which effective degree of freedom is $5$. Under the action of radiation, each molecule splits into two atoms. If the ratio of the number of dissociated molecules to the total number of molecules is $\alpha$, then find the ratio of molar specific heat capacities $(\gamma =\frac{C_P}{C_V})$.

A weightless ladder, 20 ft long rests against a frictionless wall at an angle of 60⁰ with the horizontal. A 150 pound main is 4 ft from the top of the ladder. A horizontal force is needed to prevent it from slipping choose the correct magnitude from the following

The instantaneous velocity of a particle moving in a straight line is given as $v=\alpha t+\beta t^2$, where $\alpha$ and $\beta$ are constants. The distance travelled by the particle between $1\text{s}$ and $2\text{s}$ is

Comprehension Type: A uniform thin cylindrical disk of mass M ad radius R is attached to two identical mass less springs of springs constant k which are fixed to the wall as show in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance d from its center. The axle is mass less and both the springs and the axle are in horizontal plane. The un-stretched length of each springs is L. The disk is initially at its equilibrium position with its center of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity $\overrightarrow{(}V{)}_0=V_0\hat{i}$ The coefficient of friction is $\mu$.

The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is

A moth is at a distance of $10cm$ in front of a plane mirror. An observer is standing at a distance of $30cm$ in front of the mirror. Calculate the distance between the observer and the position of the moth's image.

When the ratio of the minimum velocity to the maximum velocity is $\frac{1}{2}$ in a projectile motion, the angle of projection of projectile is?

For the planet-sun system identify the correct statement

White light is passed through a prism of angle $5^{\circ }$. If the refractive indices for red and violet colours are 1.641 and 1.659 respectively, the angle of dispersion between them is

An insect is moving on a rectangular plane from one vertex point to other along the diagonal, How many minimum number of independent informations required to completely specify its position.

Two balls $A$ and $B$ of masses $2\text{kg}$ each are moving with speeds $1\text{m/s}$ and $2\text{m/s}$ on a frictionless surface. After colliding, ball $A$ returns back with speed $0.5\text{m/s}$, then the maximum potential energy of deformation is

Two protons of masses $1.6\times {10}^{-27}\text{kg}$, goes round in a concentric circular orbits of radii $0.50\text{nm}$ and $1\text{nm}$ experience centripetal forces in the ratio of $1:4$, then the ratio of frequency of revolution of the protons is (Assume no force of interaction between protons )

If a negatively charged body comes in contact with a neutral body and 10 electrons flow towards the neutral body then find the charge acquired by the neutral body.

When a ball is thrown up vertically with velocity $V_0,$ it reaches a maximum height of '$h$'. If one wishes to triple the maximum height then the ball should be thrown with velocity. (AIIMS 2005)

Figure shows a block $A$ of mass $6m$ having a smooth semicircular groove of radius $a$ placed on a smooth horizontal surface. A block $B$ of mass $m$ is released from rest from horizontal position as shown in the figure. Find the speed of the bigger block $6m$, when the smaller block reaches the bottommost position of the groove. Take $g=10{\text{m/s}}^2$.

Which of the following combination should be selected for better tuning of an LCR circuit used for communication?

A plane wave of monochromatic light falls normally on a uniform thin film of oil which covers a glass plate. The wavelength of sources can be varied continuously. Complete destructive refraction is observed for $\lambda =5000\dot{A}$ and $\lambda =7000\dot{A}$ and for no other wavelength in between. If $\mu$ of oil is 1.3 and that of glass is 1.5, the thickness of the film will be

The second overtone of an open organ pipe has the same frequency as the first overtone of a closed pipe $L$ metre long. The length of the open pipe will be

Given below are the equations of motion of four particles $A,B,C$ and $D$.

$x_A=6t-3$;

$x_B=4t^2-2t+3$;

$x_C=3t^3-2t^2+t-7$;

$x_D=7\cos 60°-3\sin 30°$

Which of these four particles move with uniform non-zero acceleration?

Which layer of eye is transparent and thin?

A ball of $2\text{kg}$ strikes a wall with a velocity $1\text{m/s}$ at an angle ${30}^{\circ }$ with the horizontal as shown in the figure below. It reflects at the same angle from the wall in $0.1\text{s}$. Force exerted on the ball will be:

A force of $30\text{N}$ is sufficient just to pull a block of mass $4\text{kg}$, horizontally over a flat rough surface. The angle of friction is

The coefficient of friction between the block and the surface is $0.4$ in Fig. (i-iv). Match Column I with Column II.



$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{i. Force of friction is zero in}& \text{a. Fig. (i)}\\ \text{ii. Force of friction is}2.5\text{N in}& \text{b. Fig. (ii)}\\ \text{iii. Acceleration of the block is zero in}& \text{c. Fig. (iiii)}\\ \text{iv. Normal force is not equal to}2\text{g in}& \text{d. Fig. (iv)}\end{array}$

A parallel-plate capacitor has a dielectric slab in it. The slab just fills the space inside the capacitor. The capacitor is charged by a battery and then the battery is disconnected. Now, the slab is pulled out slowly at $t=0$. If at time $t$, the capacitance of the capacitor is $C$ and potential difference between the plates of the capacitor is $V$, then which of the following graphs is/are correct.

Find the value of $|\overrightarrow{a}\times \overrightarrow{b}{|}^2.$

A uniform rope of length $L$ and mass $M$ is hanging from a rigid support. The tension in the rope at a distance $x$ from the rigid support is

An ideal gas having initial pressure $P$ volume $V$ and temperature $T$ is allowed to expand adiabatically until its volume becomes $5.66V$, while its temperature falls to $T/2$. Find the number of degrees of freedom of the gas molecule.

(Take $(5.66{)}^{0.4}=2)$

Two satellites $S_1$ and $S_2$ revolve round a planet in coplanar circular orbits in the same sense. Their periods of revolution are 1 hour and 8 hours respectively. The radius of the orbit of $S_1$ is ${10}^4\text{km}$.



When $S_2$ is closest to $S_1$, find the speed of $S_2$ relative to $S_1$(in $kmph$)

In the adjoining diagram, a wavefront AB, moving in air is incident on a plane glass surface XY. Its position CD after refraction through a glass slab is shown also along with the normals drawn at A and D. The refractive index of glass with respect to air (μ=1 ) will be equal to

A metal plate of mass $200gm$ is balanced in mid–air by throwing $40$ balls/sec, each of mass $2gm$, vertically upwards from below. The balls rebound with the same speed with which they strike the plate. Find the velocity with which the balls strike the plate. (Take $g=10m/s^2$)

Water is filled in a tank upto $4\text{m}$ height. The base of the tank is at height $1\text{m}$ above the ground. At what height should the hole be made from the ground, so that the water can be sprayed upto maximum horizontal distance on ground?

Boyle's law holds for an ideal gas during

A $20\text{kg}$ wagon originally at rest, is pushed with a force of $7\text{N}$ for $1.5\text{s}$, then with a force of $5\text{N}$ for $1.7\text{s}$ and then with a force of $10\text{N}$ for $3\text{s}$. Find the final velocity of the cart.