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The manifestation of band structure in solids is due to

An airplane flying $490\text{m}$ above ground level at $100\text{m/s}$, releases a bomb. The horizontal distance it has to travel before it hits the ground is (Take $g=9.8{\text{m/s}}^2$)

Find out the phase constant $\varphi$ [in $\text{rad}$] of a particle performing SHM, if the initial conditions are as shown in the figure.

The direction of $\overrightarrow{A}\times \overrightarrow{B}$ for the vectors shown in the figure is

A person A moving in a car with a speed of $10\text{m/s}$ drops a stone of mass $10\text{gm}$. A person B on the ground measures the momentum of the stone 1 sec after the stone is dropped. Find the momentum ($\text{kg m/s}$) of the stone as measured by B. (Take $g=10{\text{m/s}}^2$)

The rate of diffusion of two gases $A$ and $B$ is in the ratio of $1:2$ and the $B$ and $C$ in the ratio $1:8$, the rate of diffusion of $C$ with respect to $A$ is

Which of the following will change their time period as they are taken to moon?

Two coils have a mutual inductance 0.005 H. The current changes in the first coil according to equation $I=I_o$sin ωt, where $I_o$= 10A and ω = 100$\Pi$ radian/sec. The maximum value of e.m.f. in the second coil is

A ball is in a hot air balloon that starts from rest at t = 0 and accelerates upwards at 2m/s². After 10s, the ball is dropped from the balloon

Draw the a-t graph of the ball from t =0

Two trains each having a speed of $30\text{km/h}$ are headed at each other on the same straight track. A bird that can fly at $60\text{km/h}$ flies off from one train when they are $60\text{km}$ apart and heads directly for the other train. On reaching the other train, it flies directly back to the first and so forth. How many trips $\left(n\right)$ can the bird make from one train to the other before the trains crash. Find the total distance that the bird travels assuming bird to be a point object.

A block of mass $5\text{kg}$ is on a rough horizontal surface at rest. Now a force($F$) of $24\text{N}$ is imparted to it with negligible impulse. If the coefficient of kinetic friction is $0.4$ and $g=9.8{\text{m/s}}^2$, then the acceleration of the block is

A block kept on a rough inclined plane as shown in the figure, remains at rest upto a maximum force $2N$ down the inclined plane. The maximum external force up the inclined plane that does not move the block is $10N$. The coefficient of static friction between the block and the plane is

[Take $g=10m/s^2]$

The ratio of kinetic energy of two bodies is $2:1$ and their angular momentum are in the ratio of $1:2$. Then the ratio of their moment of inertia will be (assume pure rotational motion)

The figure shown below represents a particle executing SHM. If the particle starts from the extreme position, $\left(P\right)$, find the correct equation of motion of particle.

How much heat must be removed by a refrigerator from $4\text{kg}$ of water at ${70}^{\circ }\text{C}$ to convert it to ice cube at $-{10}^{\circ }\text{C}$ ?

[Take specific heat of water $C_w=4200{\text{J/kg/}}^{\circ }\text{C}$, Latent heat of fusion of ice$L_f=334000\text{J/kg}$, Specific heat of ice $C_i=2100\text{J/kg K}$]

A rod is placed inside a hollow spherical shell as shown in figure. Friction acts between the rod and the shell. The number of forces to be shown in FBD of rod is/are?

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

If a man increases his speed by $2m/s$, his kinetic energy is doubled. The original speed of the man is

Find the force of interaction between $3kg$ and $2kg$ (Assume surfaces are frictionless )

The ratio of angular speeds of minute hand and hour hand of a watch is

Three particles each of mass $1\text{kg}$ are placed at the vertices of an equilateral triangle $ABC$ of side $2\text{m}$. Find the moment of inertia of system about a line perpendicular to the plane of system and passing through middle point ($D$) of side $BC$ of the triangle.

A freight car is moving on smooth horizontal track without any external force. Rain is falling at the rate of u m/s at an angle $\theta$ with the vertical. Rain drops are collected in the car at the rate of $\mu$ kg/s. If initial mass of the car is $m_0$ and velocity $v_0$ then find its velocity after time it.

Four holes of radius $R$ each are cut from a thin square plate of side $4R$ and mass $M$. The moment of inertia of the remaining portion about $z-$ axis (out of the plane) is

A small body of mass $m=10\text{kg}$ hangs at one end of a string of length $l$, the other end of which is fixed. It is given horizontal velocity $u$ at its lowest position. The string just becomes slack, when it covers an angle ${120}^{\circ }$ with the initial position. Find the tension in the string at the point of projection. Take $g=10{\text{m/s}}^2$

A $10\text{kg}$ stone is suspended with a rope of breaking strength $30\text{kg-wt}$. The minimum time in which the stone can be raised through a height $10\text{m}$ starting from rest is (taking $g=10\text{N/kg}$)

A $10\mu F$ capacitor charged to $20V$ is connected across a $15V$ battery through a switch such that positive terminal of battery is connected to negative charged plate of capacitor. When switch is closed, work done by battery of $15V$ in long time will be

The coefficient of static friction between the block of 2 kg and the table shown in figure is $\mu$s = 0.2. what should be the maximum value of m so that the blocks do not move? Take g = 10 $m/s^2$. The string and the pulley are light and smooth.

The operating temperature of an incandescent bulb (with a tungsten filament) of power $60\text{W}$ is $3000\text{K}$. If the surface area of the filament be $25{\text{mm}}^2$. Find its emissivity $e$.$(\sigma =5.67\times {10}^{-8}\text{W}/{\text{m}}^2-{\text{K}}^4)$

If $\overrightarrow{P}.\overrightarrow{Q}=PQ,$ then the angle between $\overrightarrow{P}$ and $\overrightarrow{Q}$ is

A player caught a cricket ball of mass 150 g moving at a rate of 20 m/s. If the catching process is completed in 0.1 s, the force of the blow exerted by the ball on the hand of the player is equal to

The network shown in the figure is part of a complete circuit. If at a certain instant, the current I is 5A and it is decreasing at a rate of ${10}^{3}A{s}^{–1}$ then $V_B–V_A$ equals

Two identical spheres A and B lie on a smooth horizontal circular groove at opposite ends of a diameter. At time t = 0, A is projected along the groove and at time t = T, it impinges on sphere B. If e is the coefficient of restitution, the second impact will occur after a time equal to

A bird moves from point $A$ $(1,-2,3)$ to $B$ $(4,2,3)$. If the speed of the bird is $10\text{m/s}$, then the velocity vector of the bird is:

All plates are of same area (A). Find effective capacitance between A & B

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

The minimum tangential speed of a rotating cylinder, at which the floor below the man may be removed and the man hangs resting against the wall is: [Take $g=10{\text{m/s}}^2$]

The lowest pressure (the best vaccum) that can be created in laboratory at ${20}^{\circ }\text{C}$ is ${10}^{-10}\text{mm}$ of $\text{Hg}$. At this pressure, the number of ideal gas molecules per ${\text{cm}}^3$ will be :

A force $(2\hat{i}+3\hat{j}+\hat{k})\text{N}$ acting on a body, displaces it from $\overrightarrow{r_1}=(2\hat{i}+\hat{j}+3\hat{k})\text{m}$ to $\overrightarrow{r_2}=(8t^2\hat{i}+9t^3\hat{j}+7t\hat{k})\text{m}$. Find the instantaneous power delivered by the force at $t=1\text{sec}$.

Which of these is the formula for fining maximum horizontal range of a projectile given its initial velocity and angle of projection?

A car of mass $2000kg$ is moving with a constant speed of $10m{s}^{-1}$ on a circular path of radius $20\text{m}$ on a level road. What must be the frictional force between the car and the road so that the car does not slip on the road ?

Force versus displacement curve is shown in the diagram. Find the change in kinetic energy by the force at the end of $30\text{m}$.

Lift A is going up with velocity 20m/s and acceleration of $8m/s^2$ in the downward direction. While lift B is going down with 10m/s and acceleration of $+2m/s^2$ in upward direction. At this moment a bolt in the ceiling of lift A gets loose and falls. Height of the lift is 16 m.Find the displacement of bolt relative to B when it hits the floor of A.

Two steel balls of equal diameter are connected by a rigid bar of negligible weight as shown and are dropped in the horizontal position from height $h$ above the heavy steel and brass base plates. If the coefficient of restitution between the ball and the steel base is $0.6$ and that between the other ball and the brass base is $0.4$, the angular velocity of the bar immediately after rebound is (Assume the two impacts are simultaneous and take $g=10{\text{m/s}}^2$).

A downward force of $100\text{N}$ is applied to the small piston with a diameter of $50\text{cm}$ in the hydraulic lift system as shown in figure. What will be the upward force exerted on the large position with a diameter of $1\text{m}$?

A vessel of volume $2000c{m}^3$ contains 0.1 mole of $O_2$ and 0.2 mole of $C{O}_2$. If the temperature of the mixture is 300k, find its pressure.

A point charge produces an electric field of magnitude $5.0{\text{NC}}^{-1}$ at a distance of 40 cm from it. What is the magnitude of charge?

The unit of Stefan's constant $\sigma$ is