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The acceleration due to gravity at a height $1km$ above the earth is the same as at a depth $d$ below the surface of earth. Then

Two springs of force constants $K_1$ and $K_2$ are connected to a mass m as shown. The frequency of oscillation of the mass is f. If both $K_1$ and $K_2$ are made four times their original values, the frequency of oscillation becomes:

The velocity - time graph of a particle moving in a straight line is shown in figure. The mass of the particle is 2kg.Work done by all the forces acting on the particle in time interval between t = 0 to t = 10 s is

For a gas $\frac{R}{C_v}=0.67$. This gas is made up of molecules which are

An ideal gas is expanding such that $PT=$ constant. The coefficient of volume expansion of the gas is :

A ray of light is incident at the glass–water interface at an angle i, it emerges finally parallel to the surface of water, then the value of ${\mu }_g$ would be

A person $1.6\text{m}$ tall is standing at the centre between two walls $3\text{m}$ high. What is the minimum size of the plane mirror fixed on the wall in front of him, if the person wants to see the image of the full height of the wall behind him?

A ray of light incidents on a plane mirror at an angle ${30}^{\circ }$, calculate the angular deviation of the reflected ray.

The trajectory of a particle in a vertical plane is $4y=3x-\frac{x^2}{4}$, $x$ and $y$ are the horizontal and vertical distances of the projectile from the point of projection. The angle of projection from the horizontal and the maximum height attained by the particle are

If the energy stored in spring '$\text{A}$' is $20\text{J}$, then energy stored in spring '$\text{B}$' is (under the same stretching force)

A 500 kg rocket can be accelerated at a constant rate from rest to 1800 km/h in 2 sec. What is the magnitude of the required net force?

A line passes from $(3,4)$ having $y$-intercept $c=2$. Find the slope of the line.

What do we mean when we say we completely understand mechanics?

Burst the bubbles which can be considered as the expression for Kinetic Energy.

Evaluate $\int e^x\sin xdx$

A light and a heavy body have equal momenta. Which one has greater K.E

Two discs of radii $5\text{m}$ and $3\text{m}$ are joined together along their circumference as shown in the figure below. Find the centre of mass of the system of two discs

A horizontal force $F$ pulls a ring of mass $m_1$ such that $\theta$ remains constant with time. The ring is constrained to move along a smooth rigid horizontal wire. A bob of mass $m_2$ hangs from $m_1$ by an inextensible light string. Then match the entries of Column I with that of Column II



$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{i.}& F& \text{a.}& (m_1+m_2)g\\ \text{ii.}& \text{Force acting on}{m}_2\text{is}& \text{b.}& m_{2}g\sec \theta \\ \text{iii.}& \text{Tension in the string is}& \text{c.}& m_2\frac{F}{m_1+m_2}\\ \text{iv.}& \text{Force acting on}{m}_1\text{by the wire is}& \text{d.}& (m_1+m_2)g\tan \theta \end{array}$

If in a photoelectric experiment, the wavelength of incident radiation is reduced from 6000 $A^{\circ }$ to 4000 $A^{\circ }$ then

The hero of a stunt film fires $50\text{gm}$ bullets from a machine gun, each at a speed of $1.0\text{km/s}$. If he fires $20$ bullets in $4\text{s}$, what average force does he exert against the machine gun during this period?

Block $A$ of mass $m$ is hanging from a vertical spring having stiffness $k$ and is at rest. Block $B$ of identical mass strikes the block $A$ with velocity $v$ and sticks to it. Then the value of $v$ for which the spring just attains natural lengths is

A particle of mass $m$ just completes the vertical circular motion. What will be the difference in tension at the lowest and highest point?

The surface energy of a drop of water of radius $r$ is proportional to:

The position coordinates of a projectile projected from ground on a certain planet (with no atmosphere) are given by $y=(4t-2t^2)m$ and $x=\left(3t\right)m$, where $t$ is in second and point of projection is taken as origin. The angle of projection of projectile with vertical is

In the figure shown, the ratio of the force exerted by $1\text{kg}$ body on $2\text{kg}$ body in the two cases is

The radius of gyration of a uniform rod of length $L$, about an axis passing through its centre of mass and perpendicular to it's length is:

A $20\text{kg}$ block is initially at rest on a rough horizontal surface. A horizontal force of $75\text{N}$ is required to set the block in motion. After it is in motion, a horizontal force of $60\text{N}$ is required to keep the block moving with constant speed. The coefficient of static friction is

An object of mass $50\text{kg}$ is raised to a height of $10\text{m}$ above the ground. If the object is allowed to free fall. What is kinetic energy, when it is half way down? ($g=10{\text{m/s}}^2$)

The torque acting on a dipole of moment $\overrightarrow{P}$ in an electric field $\overrightarrow{E}$ is

Two spheres each of mass $M$ and radius $\frac{R}{2}$ are connected with a massless rod of length $R$ as shown in the figure. The moment of inertia of the system about an axis $\left(Y{Y}^{\prime }\right)$ passing through the centre of one of the spheres and perpendicular to the rod is

A solid uniform cylinder of mass $M$ and radius $R$ is pivoted at its centre free to rotate about horizontal axis. A massless inextensible string is wrapped around it, and attached to a block of mass $m$ which is initially at a height $h$ above the floor. The acceleration due to gravity is $g$, directed downward. The block is released from rest. By what total angle $\theta$ does it rotate (in radians)?

A man walks on a straight road from his home to a market 2.5 km away with a speed of 5 km/h. Finding the market closed, he instantly turns and walks back home with a speed of 7.5 km/h. If the total time of travel is 40 min find the average speed of the man for the whole journey.

A projectile takes off with an initial velocity of $20\text{m/s}$ at an angle of elevation of ${45}^{\circ }$. It is just able to clear two hurdles of height $4\text{m}$ each, separated from each other by a distance $d$. What is the value of $d$? (Take $g=10{\text{m/s}}^2$)

The speed of a wave in a certain medium is 960 m/s. If 3600 waves pass over a certain point of the medium in 1 min, the wavelength is

Two Carnot engines $A$ and $B$ are operated in series. The first engine, $A$, receives heat at $800\text{K}$and rejects to a reservoir at temperature $T$. The second engine $B$ receives the heat rejected by the first engine and in turn rejects to a heat reservoir at $300\text{K}$. Find the value of $T$, when the work outputs of the engines are equal.

A $200\text{kg}$ cannon at rest contains a $10\text{kg}$ cannon ball. When fired, the cannon ball leaves the cannon with a speed of $90\text{m/s}$. What is the recoil speed of the cannon?

Two blocks of masses $3\text{kg}$and $6\text{kg}$ respectively are placed on a smooth horizontal surface. They are connected by a light spring of force constant $k=200\text{N/m}$. Initially the spring is unstretched. The indicated velocities are imparted to the blocks. Find the maximum extension of the spring.

An electron in Bohr's hydrogen atom has an energy of $-3.4eV$. The angular momentum of the electron is

Dimensional formula of resolving power of telescope and resolving power of microscope respectively are

Which of the given relations is correct?

$(c$ is speed of light, $E$ is electric field strength and $B$ is magnetic field strength$)$

A ball of mass $m$ moving with speed $u$ undergoes a head on collision with a stationary ball of mass $3m$. If collision is perfectly inelastic, find the loss of kinetic energy.

A stone is dropped into a river from a bridge 125 m above the water. Another stone is thrown vertically down 1.00s after the first is dropped. The stones strike the water at the same time.

Which of these is the correct velocity time graph of both the stones?

The magnitude of components of a vector $\overrightarrow{a}$ along x and y axis respectively are

Find the relation between the acceleration of sphere $\left(a\right)$ and acceleration of wedge $\left(b\right)$ in the figure shown

A person throws vertically $10$ balls per second with the same velocity. He throws a ball whenever the previous one is at its highest point. The height to which the balls rise is

An object is approaching a fixed plane mirror with velocity $10\text{m/s}$ making an angle of ${60}^{\circ }$ with the normal to the mirror. The speed of image with respect to the mirror is

The initial velocity of a particle is 10 m/sec and its retardation is 2 m/$se{c}^2$. The distance moved by the particle in $5^{th}$ second of its motion is

A particle is projected vertically upwards with speed $20\text{m/s}$ from top of a tower of height $20\text{m}$ as shown in figure. Given $B$ is top most point of trajectory and $C$ is at same height as $A$ :

$\begin{array}{cccc}& \text{Column - I}& & \text{Column - II}\\ \left(A\right)& \text{ratio of maximum height from ground (BD)}& \left(P\right)& \frac{1}{\sqrt{2}}\\ & \text{to the initial height from ground (AD)is}& & \\ \left(B\right)& \text{ratio of distance travelled in 1st second to}& \left(Q\right)& \frac{1}{\sqrt{3}}\\ & \text{the distance travelled in 2nd second is}& & \\ \left(C\right)& \text{ratio of initial speed at A to the final}& \left(R\right)& 1\\ & \text{just before reaching to ground (D) is}& & \\ \left(D\right)& \text{ratio of time taken from A to C and time}& \left(S\right)& 2\\ & \text{taken from A to B is}& & \\ & & \left(T\right)& 4\\ & & \left(U\right)& 3\end{array}$

Which of the following option has the incorrect combination considering column-I and column-II

In the figure shown, the coefficient of friction between all surfaces in contact is $0.4$. Taking $g=10m/s^2$, find the tension $T$ in the string.