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Mass density and radius of a solid sphere are ρ, R respectively. Gravitational field at an internal point which is at a distance r from the centre is.

Two rods A and B of different materials are welded together as shown in figure. Their thermal conductivities are $K_1$ and $K_2$. The thermal conductivity of the composite rod will be

Two balls of masses $2\text{kg}$ and $4\text{kg}$ after the collision are moving with speeds $4\text{m/s}$ and $2\text{m/s}$ as shown in figure. Find the impulse of reformation for the $4\text{kg}$ ball.

A uniform metre scale of length 1 m is balanced on a fixed semi – circular cylinder of radius 30 cm as shown in figure. One end of the scale is slightly depressed and released. The time period (in seconds) of the resulting simple harmonic motion is $(Takeg=10m{s}^{-2})$

Assuming that the atmosphere has the same density anywhere as at sea level $(\rho =1.3{\text{kg/m}}^3)$ and $g$ to be constant $(g=10{\text{m/s}}^2)$, what should be the approximate height of the atmosphere so that the atmospheric pressure at sea level is $p_0=1.01\times {10}^5{\text{N/m}}^2$ ?

A body released from a height h takes time t to reach earth’s surface. The time taken by the same body released from the same height to reach the moon’s surface is

Two particles $A$ and $B$ initially separated by distance $X$ are projected simultaneously in the directions shown in figure with speed $v_A=20\text{m/s}$ and $v_B=10\text{m/s}$ respectively. They collide in air after $\frac{1}{2}\text{s}.$ Find the distance $X$.

A massless platform is kept on a light elastic spring, as shown in the figure. When a particle of mass $0.1\text{kg}$ is dropped on the pan from a height of $0.24\text{m}$, the spring gets compressed by $0.01\text{m}$. From what height should the same particle be dropped to cause a compression of $0.04\text{m}$ in the spring?

Neglect the effect of collision.

Instantaneous velocity of a particle moving in a straight line is given as $v=(4+4\sqrt{t})$$\text{m/s}$, for the first $5\text{sec}$ of its motion, there after velocity becomes constant. Find the acceleration and displacement of the particle at time $t=3.0\text{sec}$.

A projectile is thrown with a speed of $100m/s$ making an angle of ${60}^{\circ }$ with the horizontal. Find the minimum time after which its inclination with the horizontal is ${45}^{\circ }$.

From above graph choose correct option for $t_1$ to $t_2$ where $t_1$ is the time at which velocity of particle = 0 $t_2$ is the time at which velocity of particle is negative.

Find the current through 4 $\Omega$ resistor in the circuit shown.

A uniform rod of mass $m$ and length $l$ can rotate in a vertical plane about a smooth horizontal axis hinged at point $H$. Find the force exerted by the hinge just after the rod is released as shown in the figure.

The monkey $\text{B}$ shown in the figure is holding onto the tail of the monkey $\text{A}$ which is climbing up a rope. The masses of monkeys $\text{A}$ and $\text{B}$ are $5\text{kg}$ and $2\text{kg}$ respectively. If $\text{A}$ can tolerate a tension of $30\text{N}$ in its tail, what force should it apply on the rope in order to carry the monkey $\text{B}$ with it ? (take $g=10{\text{m/s}}^2$)

A bird flies for 4 s with a velocity of $|t-2|$ m/s in a straight line, where t is time in seconds. It covers a distance of

An experiment takes $10\text{min}$ for an electric kettle to heat a certain quantity of water from $0^{\circ }\text{C}$ to ${100}^{\circ }\text{C}$. If it takes $54\text{min}$ to convert this water at ${100}^{\circ }\text{C}$ into steam, then latent heat of steam is:

A satellite can be in a geostationary orbit around earth at a distance $r$ from the center. If the angular velocity of earth about its axis doubles, a satellite can now be in a geostationary orbit around earth, if its distance from the center is

Two ice skaters A and B approach each other at right angles. Skater A has a mass $30\text{kg}$ and velocity $1\text{m/s}$ skater B has a mass $20\text{kg}$ and velocity $2\text{m/s}$. They meet and cling together. The final velocity of the couple is

If the direction of the initial velocity of the charged particle is neither along nor perpendicular to that of the magnetic field, then the orbit will be

When the electrical conductivity of a semi- conductor is due to the breaking of its covalent bonds, then the semiconductor is said to be

A point source of light is used in a photoelectric effect. If the source is removed farther from the emitting metal, the stopping potential.

A piston of cross-sectional area $100{\text{cm}}^2$ is used in a hydraulic pump to exert a force of ${10}^7\text{dyne}$ on the water. The cross-sectional area of the other piston which supports a truck of mass $2000\text{kg}$ is

Two particles of equal mass have co-ordinates $(2,4,6)\text{m}$ and $(6,2,8)\text{m}$. One particle has a velocity ${\overrightarrow{v}}_1=\left(2\hat{i}\right)\text{m/s}$ and another particle has velocity ${\overrightarrow{v}}_2=\left(2\hat{j}\right)\text{m/s}$ at time $t=0$. The coordinates of the centre of mass of the system at time $t=1\text{s}$ will be

A cylinder of capacity $20\text{L}$ is filled with $H_2$ gas. The total average kinetic energy of translatory motion of its molecules is $1.5\times {10}^5\text{J}$. The pressure of hydrogen in the cylinder is

A siren producing a sound of frequency $750\text{Hz}$ moves away from an observer at rest towards a cliff at a speed of $30\text{m/s}$. The frequency of echo heard by the observer reflected from the cliff is

[speed of sound in air $=330\text{m/s})$

Four identical rods of mass $M=6\text{kg}$ each are welded at their ends to form a square and then welded to a massive ring having mass $M=4\text{kg}$ and radius $R=1\text{m}$. If the system is allowed to roll down the incline of inclination $\theta ={30}^{\circ }$.





The minimum value of friction coefficient to prevent slipping is

For the given oscillatory motion, $F=-k(x-b{)}^3$, find out the mean position of the particle.

A standing wave $y=\text{A sin}\left(\frac{20\pi x}{3}\right)\text{cos (1000}\pi \text{t)}$ is set up in a taut string, where $x$ and $y$ are in metre and $t$ is in seconds. The distance between first two successive points oscillating with the amplitude $\frac{A}{2}$, from the fixed end will be equal to

Weightlessness experienced while orbiting the Earth in International Space Station, is the result of

Two circular discs A and B are of equal masses and thickness but made of metals with densities $d_A$ and $d_B(d_A>d_B).$ If their moments of inertia about an axis passing through centres and normal to the circular faces be $I_A$ and $I_B$, then

Two thin converging lenses are placed on a common axis, so that the center of one of them coincides with the focus of the other. An object is placed at a distance twice the focal length from the left-hand lens. Where will its image be? What is the lateral magnification? The focal length of each lens is $f$.

For which combination of temperatures the efficiency of Carnot's engine is highest?

If a converging lens of focal length f is cut into two parts and kept as shown in figure, the effective focal length of the combination will be ___

There are two point charges of +1C each separated by 10m in vacuum. Calculate the change in potential energy of the system if they are brought to a separation of 5m.

To an ideal non-linear triatomic gas, $800\text{cal}$ of heat energy is given at constant pressure. If vibrational modes are neglected, then the energy used by the gas in doing work against the surroundings is

Water flows through a frictionless duct with a cross-section varying as shown in the figure. The pressure $P$ at points along the axis (starting from left end to right end) is represented by

A boy whirls a stone in a horizontal circle at a height of $31.25\text{m}$ above ground level and radius of circle being $3\text{m}$. Somehow, the string breaks and stone flies off horizontally and strikes the ground after travelling horizontal distance of $5\text{m}$. Find the magnitude of centripetal acceleration of stone while being in circular motion. Assume stone to be in uniform circular motion.

Two plane mirrors are inclined at an angle $\theta$ as shown in the figure. Find the value of $\theta$ such that the light ray retraces its path after second reflection.

A wire of area of cross-section $3.0m{m}^2$ and natural length $50cm$ is fixed at one end and a mass of $2.1kg$ is hung from the other end. Find the elastic potential energy stored in the wire in steady state. Young’s modulus of the material of the wire $=1.9\times {10}^{11}N{m}^{-2}$. Take $g=10m{s}^{-2}$.

If $F_1=10\text{N}$ and $F_2=20\text{N}$ as shown in the figure, the resultant $R$ will be [Hint: $\cos {65}^{\circ }=0.423$]

A bullet of mass $m_2$ is fired from a gun of mass $m_1$ with horizontal velocity $v$. A plane mirror is fixed to the gun facing towards the bullet. The velocity of the image of the bullet formed by the plane mirror with respect to the bullet is (Neglect the mass of mirror)

The value of acceleration due to gravity at a height $h=\frac{R}{4}$ from the surface of earth is



[Where R is radius of earth and g is the acceleration due to gravity at earth’s surface]



पृथ्वी की सतह से ऊँचाई $h=\frac{R}{4}$ पर गुरुत्वीय त्वरण का मान है



[जहाँ R पृथ्वी की त्रिज्या है तथा g पृथ्वी की सतह पर गुरुत्वीय त्वरण है]

The springs in fig. $A$ and $B$ are identical but the length of spring $A$ is three times the length of each spring in $B$. The ratio of period $T_A/T_B$ is

One $\text{mole}$ of an ideal diatomic gas at $300\text{K}$ is compressed adiabatically to one-fourth of its volume. If $\gamma =1.5$, the work done on gas will be

The coordinates of a particle moving in a plane are given by x(t) = a cos(pt) and y(t) = a sin (pt)

where a, and p are positive contants of appropriate dimensions. Then,

A gas mixture of $2$ moles of $O_2$ and $4$ moles of $Ar$ are at temperature $T$. Neglecting all the vibrational modes, the total internal energy of the system is