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The smallest velocity (in $m/s$) with which a particle may be projected in order to have a range of $40m$ on the horizontal plane is

(Take $g=10m/s^2$)

The two opposite faces of a cubical piece of iron (thermal conductivity $=0.2\text{CGS units}$) are at ${100}^{\circ }\text{C}$ and $0^{\circ }\text{C}$. The face at $0^{\circ }\text{C}$ is in contact with the ice at $0^{\circ }\text{C}$. If the area of a surface is $4{\text{cm}}^2$ and latent heat of fusion of ice is $80\text{J/g}$, then the mass of ice melted in $10$ minutes will be:

A heavy uniform chain lies on a horizontal table top. If the coefficient of friction between the chain and the table surface is $0.25$, then the maximum fraction of length of chain that can overhang on the edge of the table is

A point source of light $S$ is placed at a distance $L$ in front of the centre of plane mirror of length$\left(d\right)$. A man walks in front of the mirror along a line parallel to the mirror, at a distance $2L$ as shown below. The distance over which the man can see the image of the light source in the mirror is

A steel cable 3.00$c{m}^2$ in cross-sectional area has a mass of 2.40 kg per meter of length.; If 500m of the cable is hung over a vertical cliff, how much does the cable stretch under its own weight? $(Y_{steel}=2.00\times {10}^{11}N{m}^{-2};g=10m{s}^{-2})$

The maximum and minimum magnitudes of the resultant of two vectors are $23$ units and $7$ units respectively. If these two vectors are acting at right angles to each other, the magnitude of their resultant will be

In YDSE, find the thickness of a glass slab $(\mu =1.5)$ which should be placed before the upper slit $S_1$ so that the central maximum now lies at a point where $5^{th}$ bright fringe was laying earlier (before inserting the slab). Wavelength of light used is $5000\dot{A}$.

A solid cube is placed on a horizontal surface. The coefficient of friction between them is $\mu$, where $\mu <1/2$. A variable horizontal force is applied on the cube's upper face, perpendicular to one edge and passing through the mid-point of the edge, as shown in figure. The maximum acceleration with which it can move without toppling is

A flatcar of mass $m_0=20\text{kg}$ is moving to the right due to a constant horizontal force $F=10\text{N}$. Sand spills on the flatcar from a stationary hopper. The velocity of loading is constant and equal to $\mu =1\text{kg/s}$. Find the time dependence of the velocity of the flatcar in the process of loading. The friction is negligibly small.

At absolute zero temperature, pressure of a gas will be

A mass M is split into two parts, m and (M-m), which are then separated by a certain distance. What ratio of $\frac{m}{M}$ maximizes the gravitational force between the two parts

A boat is moving with a velocity $3\hat{i}-5\hat{j}$ with respect to ground. The water in the river is moving with a velocity $-\hat{i}+2\hat{j}$ with respect to ground. The relative velocity of the boat with respect to water is

When does the body come back to its starting point?

A metallic sphere floats in an immiscible mixture of water and a liquid such that its $\frac{4}{5}\text{th}$ volume is in water and $\frac{1}{5}\text{th}$ volume is in the liquid. Then, density of the metal is

A heavy steel ball of mass greater than 1 kg moving with a speed of 2 $mse{c}^{-1}$collides head on with a stationary ping-pong ball of mass less than 0.1 gm. The collision is elastic. After the collision the ping-pong ball moves approximately with speed

Electric field vs. distance along the axis is shown for a uniformly charged circular ring

Which of the following represents the complete graph?

Five persons $A,B,C,D$ and $E$ are pulling a cart of mass $100\text{kg}$ on a smooth surface and cart is moving with acceleration $3{\text{m/s}}^2$ in east direction. When person $A$ stops pulling, it moves with acceleration $1{\text{m/s}}^2$ in the west direction. When person $B$ stops pulling, it moves with acceleration $24{\text{m/s}}^2$ in the north direction. The magnitude of acceleration of the cart when only $A$ and $B$ pull the cart keeping their directions same as the old directions, is

The masses of $10\text{kg}$ and $20\text{kg}$ respectively are connected by a massless spring in fig. A force of $200$ newton acts on the $20\text{kg}$ mass. At the instant shown, the $10\text{kg}$ mass has acceleration $12{\text{m/sec}}^2$. What is the acceleration of $20\text{kg}$ mass?

A rigid cubical block A of mass M and side L is fixed rigidly on to another cubical block of the same dimensions and of modulus of rigidity $n$ such that the lower face of A completely covers the upper face of B. the lower face of B is rigidly held on a horizontal surface. A small F is applied perpendicular to one of the side faces of A.After the force is withdrawn, block A executes small oscillations, the time period of which is

Complete the following formulae.

A gun moving at a speed of $30m/s$ fires a bullet (in the same direction gun is moving) at an angle ${30}^o$ with the horizontal and a velocity of $150m/s$ relative to the gun. Find the distance between the gun and the bullet when the bullet hits the ground. (Take $g=10m/s^2$)

The wavelength of light visible to eye is of the order of

Three objects $A$ (solid sphere), $B$ (thin circular disk) and $C$ (circular ring) each have the same mass $M$ and radius $R$. They all spin with the same angular speed $\omega$ about their own symmetry axes. The amounts of work ($W$) required to bring them to rest, would satisfy the relation

The internal energy of a gas is given by $U=(2PV+20)\text{J}$. It expands from $100{\text{cm}}^3$ to $200{\text{cm}}^3$ against a constant pressure of $200\text{kPa}$. Calculate the heat absorbed by the gas in the process.

Whenyou are pedalling a bicycle, the force of friction exerted

by the ground on the two wheels is such that it acts

In a YDSE performed with wavelength $\lambda =5890\dot{A}$, the angular fringe width is ${0.40}^{\circ }$. What is the angular fringe width if the entire set - up is immersed in water ? $(\mu =\frac{4}{3})$

Density of wood is $0.5\text{g/cc}$ in the $CGS$ system of units. The numerical value of density in $MKS$ system of units is

A coin is pushed down tangentially from an angular position $\theta$ on a circular surface, with velocity $v$ as shown. If the cofficient of friction between the coin and surface is $\mu$, find the tangential acceleration of the coin.

A stone is released from the top of the tower of height $20\text{m}$. Then its final velocity just before touching the ground will be

An ant starts from the origin and walks along a straight line. After every second, its position (in mm) is observed by a curious 5-year old. These are his readings:

Position (mm) 0 10 40 100 200

After time (s) 0 1 2 3 4

By looking at this table, which of the following could be a possible description of the ant's motion is

(a) Uniform velocity

(b) uniform acceleration (with non-uniform velocity)

(c) Uniform jerk (with non-uniform accelerator)

(d) cannot be determined (can be any of the above)

The elevator shown in the figure is descending with an acceleration of $2m/s^2$. The mass of the block $A=0.5\text{kg}$. The force exerted by the block $A$ on the block $B$ is (take $g=10{\text{m/s}}^2$)

The truth table for the following logic circuit is:

A circular disc X of radius 'R' is made from an iron plate of thickness 't', and another disc Y of radius '4R' is made from an iron plate of thickness '$\frac{t}{4}$'. Then the relation between the moment of inertia $I_x$ and $I_y$ is

A vertical bar magnet is dropped from position on the axis of a fixed metallic coil as shown in fig-I. In fig-II the magnet is fixed and horizontal coil is dropped. The acceleration of the magnet and coil are $a_1$ and $a_2$ respectively then

A projectile is projected from the origin in the $x-y$ plane with a velocity of $60\text{m/s}$ at a ${30}^{\circ }$ with the horizontal. Find the position vector of the projectile after $2\text{s}$. Take $(g=10{\text{m/s}}^2)$

In a gravity free space, a spacecraft of mass $10\text{kg}$, having $20\text{kg}$ of fuel is initially at rest. The fuel is ignited and the exhaust velocity is $0.1\text{km/s}$. Then, speed gained by the spacecraft when the fuel is fully consumed is

The resistance of a platinum resistance thermometer of ice point, the steam point and the boiling point of sulphur are 2.50, 3.50 and 6.50$\Omega$ respectively. Find the boiling point of sulphur on the platinum scale. The ice point and steam point measure $0^{\circ }$ and ${100}^{\circ }$ respectively.

A boy walks to his school at a distance of $9km$ with constant speed of $3km{h}^{-1}$ and walks back with a constant speed of $6km{h}^{-1}$. His average speed for round trip in ($km{h}^{-1}$) is

Range of frequencies allotted for commercial FM radio broadcast is

A black body at a temperature of ${327}^{\circ }\text{C}$ radiates $4{\text{cal cm}}^{-2}{\text{s}}^{-1}$ heat per unit area per unit time. At a temperature of ${927}^{\circ }\text{C}$, the rate of heat radiated per unit area per unit time in ${\text{cal cm}}^{-2}{\text{s}}^{-1}$ will be

The faintest sound that can be heard has a pressure amplitude of about $4\times {10}^{-5}{\text{N/m}}^2$ and the loudest that can be heard without pain has a pressure amplitude of about $2.8{\text{N/m}}^2$. Determine the intensity of the faintest sound (both in ${\text{W/m}}^2$ and in $\text{dB}$).

[Assume, air density is $1.3{\text{kg/m}}^3$ and velocity of sound is $330\text{m/s}$ and take ${\log }_{10}1.86=0.269$]

The acceleration of the particle moving in positive $x-$ direction with an initial velocity of $20\text{m/s}$ is $4{\text{m/s}}^2$. Its velocity at $t=5\text{sec}$ is

A thin rod of mass M and length L is bent in semicircle as shown in figure. The gravitational force on a particle with mass m at O, the centre of curvature is...

For the damped oscillator system shown with m = 250g, k = 85 N/m, and b = 70 g/s, what is the ratio of the oscillation amplitude at the end of 20 cycles to the initial oscillation amplitude?

A ball of mass $m$ moving with a speed $u$ collides elastically with another identical ball moving with velocity $u_2$. Find the angle between velocities after collision if they collide obliquely and $u_2$ = 0​

A body of mass 3 kg is under a force, which causes a displacement in it is given by $S=\frac{t^3}{3}$ (in m). Find the work done by the force in first 2 seconds

$n$ small balls, each of mass $m$, impinge elastically each second on a surface with velocity $u$. The force experienced by the surface will be