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In damped oscillations, if the amplitude becomes half of its maximum value in $1\text{s}$, then after $2\text{s}$ amplitude will be

($A_o$ is initial amplitude)

A particle executes SHM, at what value of displacement are the kinetic and potential energies equal ?

The speed of a homogeneous solid sphere after rolling down an inclined plane of vertical height $h$ from rest, without sliding is

Find the velocity of the body of mass $2kg$ from the momentum-time graph at the end of $5$ seconds.

Consider two particles $P$ and $Q$. Particle $P$ moves from $A$ to $B$ and particle $Q$ moves from $C$ to $D$. If the displacement of particle $P$ and $Q$ be $x$, $y$ respectively. Then:

What is the equivalent spring constant for the syatem shown below in the figure?

The amplitude of a wave represented by the displacement equation $y=\frac{1}{\sqrt{a}}\sin (kx-\omega t)\pm \frac{1}{\sqrt{b}}\cos (kx-\omega t)$ will be

Three S.H.M's of equal amplitude A and equal time period in the same direction combine. The difference in phase between each pair is ${60}^{\circ }$ ahead of the other. The amplitude of the resultant oscillation is :

Any radiation in the ultra violet region of Hydrogen spectrum is able to eject photoelectrons from a metal. Then the threshold frequency of the metal for a threshold $\lambda =\frac{1}{R}=911\dot{A}$,is nearly

Two infinite planes each with uniform surface charge density $+\sigma C/m^2$ are kept in such a way that the angle between them is ${30}^o$. The electric field in the region shown between them is given by:

A uniform rod of mass M and length ℓ is placed on a smooth horizontal surface with its one end pivoted to the surface. A small ball of mass m moving along the surface with a velocity v₀, perpendicular to the rod, collides elastically with the free end of the rod. Find the impulse applied by the pivot on the rod during collision. (Take M/m = 2)

A material has Poisson's ratio $0.5$. If a uniform rod made of the material suffers a longitudinal strain of $2\times {10}^{-3}$, what is the percentage increase in volume?

Find the direction of friction on $A$ and $B$ respectively at the common surface between blocks $A$ and $B$ for the figure shown.

Vector $\overrightarrow{A}$ makes equal angles with x, y and z axis. Value of its components (in terms of magnitude of $\overrightarrow{A}$) will be

A force $F=-\frac{k}{x^2}(x\ne 0)$ acts on a particle in $x-$direction. Find the work done by this force in displacing the particle from $x=+a$ to $x=+2a.$ Here $k$ is a positive constant.

Two parallel, conducting rails $1$ and $2$ are kept perpendicular to a uniform magnetic field $\left(B\right)$. The rails are at seperation $l$ and are joined at their ends by a resistance $R$. A conducting bar $AB$ is placed on the rails making ${60}^{\circ }$ with them. The bar is pulled with a velocity $v$ parallel to the rails. Find the current in the resistance $R$.

A boy running on a circular track of radius $10\text{m}$ completes $3$ revolution and comes back to the point from where he started. Calculate the angular displacement of the boy $\left(\text{in rad}\right)$.

The acceleration of a particle starting from rest varies with time according to the relation $a=kt+c$. The velocity $v$ of the particle after time $t$ will be

For a particle moving along $x$ - axis, the acceleration $a$ of the particle in terms of its $x$- coordinate $x$ is given by $a=-9x$, where $x$ is in meters and $a$ is in ${\text{m/s}}^2$. Take acceleration, velocity and displacement in positive $x$ - direction as positive. The initial velocity of particle at $x=0$ is $u=+6\text{m/s}$. The velocity of particle at $x=2\text{m}$ will be

Cream gets separated out of milk when it is churned, it is due to

Find the time period of small oscillations for the block of mass $m$ as shown in the figure. [All springs are identical]

An electromagnetic wave of wavelength ${\lambda }_0$ (in vacuum) passes from $P$ towards $Q$ crossing three different media of refractive index $\mu ,2\mu$ and $3\mu$ respectively as shown in figure. ${\varphi }_P$ and ${\varphi }_Q$ are the phase of the wave at point $P$ and $Q$. Find the phase difference ${\varphi }_Q-{\varphi }_P$ [Take: $\mu =1$]

A block with initial velocity $2\text{m/s}$ is sliding on a horizontal frictionless surface as shown in the figure. What is the value of $\theta$ when the block leaves the surface?

A uniform square plate of side $a$ is hinged at one of its corners as shown. It is suspended such that it can rotate about horizontal axis. Find out its time period for small oscillations about its equilibrium position.

The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of $x$ from it is $I\left(x\right)$. Which one of the graphs represents the variation of $I\left(x\right)$ with $x$ correctly ?

Two monochromatic sinusoidal waves, each of intensity $I$, have a constant phase difference of $\varphi$. These waves when superimposed, form a resultant sinusoidal wave. Then, the intensity of the resultant wave is

What is the distance of the body from the starting point at t = 30 s?

What is the distance of the body from the starting point at t = 30 s?

Ratio of maximum to minimum intensity during interference of two waves is $25:1$. The amplitude ratio of these two waves is

Calculate the approximate height (h) of Earth's atmosphere extending above the surface of the Earth, assuming the density of air to be uniform [$1.3kg{m}^{-3}$] and pressure at the surface of Earth to be ${10}^{5}Pa$. (Assume the Earth to be perfectly spherical and no variation in 'g' due to height).

A bullet with muzzle velocity $500\text{m/s}$ is to be shot at a target $1000\text{m}$ away in the same horizontal line. At what height above the target, a rifle must be aimed so that the bullet will hit the target ?

A cord is wound around the circumference of a bicycle wheel (without tyre) of diameter $1\text{m}$. A mass of $2\text{kg}$ is tied to the end of the cord and it s allowed to fall from rest. The weight falls $2\text{m}$ in $4\text{s}$. The axle of the wheel is horizontal and the wheel rotates with its plane vertical. The angular acceleration produced is (Take $g=10{\text{ms}}^{-2}$)

If linear density of a rod of length 3 m varies as $\lambda =1+x$, then the position of the centre of mass of the rod is

A vessel has $6\text{g}$ of hydrogen at pressure $P$ and temperature $500\text{K}$. A small hole is made in it so that hydrogen leaks out. How much hydrogen leaks out if the final pressure is $P/2$ and temperature falls to $300\text{K}$ when the hole is sealed?

Find coordinates of the centre of mass of a uniform $L-$shaped lamina (a thin flat plate) wrt origin, having dimensions as shown in figure. (Consider mass of lamina as $3mkg$)

Starting from the mean position a body oscillates simple harmonically with a period of 2 s. After what time will its kinetic energy be 75% of the total energy ?

A particle of mass $2\text{kg}$ is moving in $xy$ plane under the action of a force. Its velocity and acceleration at a certain instant of time are given by $\overrightarrow{v}=(3\hat{i}+4\hat{j})\text{m/s}$ and $\overrightarrow{a}=(5\hat{i}+4\hat{j}){\text{m/s}}^2$. Find the power delivered by the force at that instant of time.

In the following circuit, the supply voltage is defined as: V(t) = 230 sin( 314t - π/6) and L = 2.2H. Determine the value of the current flowing through the coil.

The angular velocity of a particle is given by $\omega =1.5t-3t^2+2$, the time when its angular acceleration ceases to exist will be -

A block is shot with an initial velocity of $5\text{m/s}$ on a rough horizontal plane. Find the distance covered by the block till it comes to rest. Given the coefficient of friction between the surfaces of contact is $0.2\text{and}g=10m/s^2$.

The third harmonic of a closed organ pipe is equal to the second overtone of an open organ pipe. If the length of open organ pipe is $60\text{cm}$, then the length of closed organ pipe will be

A particle moves according to the equation $x=2t^2-5t+6$. Find (i) average velocity in the first $3\text{sec}$ and (ii) velocity at $t=3\text{s}$.

Calculate the bulk modulus of air from the following data about a sound wave of wavelength 35 cm travelling in air. The pressure at a point varies between $(1.0\times {10}^{5}and14)$ Pa and the particles of the air vibrate in simple harmonic motion of amplitude $5.5\times {10}^{-6}$ m.

A body is projected with a velocity $u$ so that its horizontal range is twice the greatest height attained. The value of its range is

(Take $g=10{\text{m/s}}^2$)

The coefficient of linear expansion of brass and steel are

${\alpha }_1$ and ${\alpha }_2$. If we take a brass rod of length $l_1$ and steel rod of length $l_{2}at{0}^{\circ }C,$ their difference in length $(l_2-l_1)$will remain the same at a temperature if

The maximum range of a projectile is $22\text{m}$. When it is thrown at an angle of ${15}^{\circ }$ with the horizontal, its range will be-

A man throws a ball of mass $m$ on a vertical wall with speed $v$. If the ball bounces back with the same speed, find out the magnitude of impulse on the ball, imparted by normal reaction from the wall.