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A projectile is thrown into space so as to have maximum horizontal range R. Taking the point of projection as origin, the co-ordinates of the point where the speed of the particle is minimum are

If a given mass of gas occupies a volume of $10\text{cc}$ at $1\text{atm}$ pressure and temperature ${100}^{\circ }\text{C}$. What will be its volume at $4\text{atm}$, the temperature being the same?

A small sphere is given vertical velocity of magnitude $v_0=5{\text{ms}}^{-1}$ and it swings in a vertical plane about the end of a massless string. The angle $\theta$ with the vertical at which the string will break, given that it can withstand a maximum tension equal to twice the weight of the sphere, is

Two blocks are connected over a massless pulley as shown in the figure. The mass of block $A$ is $10\text{kg}$. If block $A$ slides down the inclined plane at constant speed, then mass of block $B$ in $\text{kg}$ is

A 5g piece of ice at $-{20}^{\circ }$C is put into 10g of water at ${30}^{\circ }$C. Assuming that heat is enchanged only between the ice and the water, find the final temperature of the mixture. Specific heat capacity of ice $=2100Jk{g}^{-1}{}^{\circ }{C}^{-1}$. Specific heat capacity of water $=4200Jk{g}^{-1}{}^{\circ }{C}^{-1}$ and latent heat of fusion of ice $3.36\times {10}^{5}Jk{g}^{-1}$.

A man of mass $50kg$ stands on a frame of mass $30kg$. He pulls on a light rope which passes over a pulley. The other end of the rope is attached to the frame. For the system to be in equilibrium, what force must the man exert on the rope? (Take $g=10m/s^2$)

A longer solenoid of diameter $0.1m$ has $2\times {10}^4$ turns per meter. At the center of the solenoid, a coil of $100$ turns and radius $0.01m$ is placed with its axis coinciding with the solenoid axis. The current in the solenoid reduces at a constant rate to $0A$ from $4A$ in $0.05s$. If the resistance of the coil is $10{\pi }^2\Omega$, the total charge flowing through the coil during this time is

The stick applies a force of $2\text{N}$ on the ring$(R=0.5\text{m})$ and the ring rolls without slipping on the ground as shown in figure. The coefficient of friction between the stick and the ring is

If the kinetic energy of a free electron doubles, its de-Broglie wavelength changes by the factor

An infinite number of charges, each equal to q, are placed along the x-axis at x =1, x=2, x=8, ... and so on. The potential at the point x =0 due to this set of charges is

Derivative of $f\left(x\right)=x^2\sin x$ is

A man of mass $M$ stands at one end of a plank of length $L$ which is at rest on a frictionless horizontal surface. The man walks to the other end of the plank. If mass of the plank is $\frac{M}{3}$, the distance that the man moves relative to the ground is

A uniform solid brass sphere is rotating with angular speed ${\omega }_0$ about a diameter. If its temperature is now increased by ${100}^{\circ }C$. What will be its new angular speed. (Given ${\alpha }_B=2.0\times {10}^{-5}{per}^{\circ }C$)

Three blocks $A,B$ and $C$ of equal masses $‘m^{\prime }$ each are placed one over the other on a frictionless table. The coefficient of friction between any two blocks is $\mu$. Find the maximum value of mass of block $D$ so that the blocks $A,B$ and $C$ move without slipping over each other.

For the given system, find the extension in the spring if $m_1=4\text{kg}$ and $m_2=6\text{kg}$. Take $g=10{\text{m/s}}^2$.

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of ${30}^{\circ }$ with each other. When suspended in a liquid of density $0.8{\text{g /cm}}^3$, the angle remains the same. If density of the marterial of the sphere is $1.6{\text{g /cm}}^3$, then dielectric constant of the liquid is

A plank of mass $m_1=8kg$ with a bar of mass $m_2=2kg$ placed on its rough surface, lie on smooth floor of elevator ascending with an acceleration $\frac{g}{4}$. The coefficient of friction is $\mu =\frac{1}{5}$ between $m_1$ and $m_2$. A horizontal force $F=30N$ is applied to the plank. Then the acceleration of bar and the plank in the reference frame of elevator are: ($g=10{\text{m/s}}^2)$

A man of mass $50\text{kg}$ is pulling a rope on a plank of mass $100\text{kg}$ kept on a smooth surface as shown, with force of $100\text{N}$. If both man and plank move together, find force of friction acting on man.

Given coefficient of friction between man and plank is $\frac{1}{6}$.

A square plate of edge ${\frac{{}^{\prime }a}{2}}^{\prime }$ is cut from a uniform square plate of edge 'a' as shown in figure. Mass of square plate of edge 'a' is $M$. Find out the moment of inertia of the remaining square plate about an axis passing through 'O' (center of square plate of side 'a') and perpendicular to the plane of the plate.

The wavelength of light in two liquids 'x' and 'y' is 3500 $\stackrel{\circ }{A}$ and 7000 $\stackrel{\circ }{A}$, then the critical angle of x relative to y will be

A rectangular loop with a sliding connector of length $10\text{cm}$ is situated in uniform magnetic field perpendicular to plane of loop. The magnetic induction is $0.1$ Tesla and resistance of connector $\left(R\right)$ is $1$ ohm. The sides $AB$ and $CD$ have resistances $2$ ohm and $3$ ohm respectively. Find the current in the connector during its motion with constant velocity $1\text{m/s}$.

A stone is dropped into water from a bridge 44.1 m above the water. Another stone is thrown vertically downward 1 sec later. Both strike the water simultaneously. What was the initial speed of the second stone?

A wire of length $2\text{m}$ is suspended from the top of a roof at one end and a load of $W$ is attached at the other end. Cross-sectional area of the wire is $5\times {10}^{-7}{\text{m}}^2$ and elongation $\left(\Delta l\right)$ is shown in the graph. Find the Young's modulus of elasticity of the wire material.

Two particles of equal masses are revolving in circular paths of radii $r_{1}and{r}_2$ respectively with the same speed. The ratio of their centripetal forces is

A photon of energy hv is absorbed by a free electron of a metal having work function $\varphi <hv$

A sphere of mass $M$ and diameter $D$ is heated by temperature $\Delta T$. If coefficient of linear expansion is $\alpha$, then find the change in the surface area of sphere.

Two blocks are connected by a string, as shown in figure. They are released from rest. The coefficient of kinetic friction between the upper block and the surface is 0.5. Assume that the pulley is massless and frictionless. Their common speed after they have moved a distance $5m$ will be: ($g=10m{s}^{-2}$)

A man of mass M stands at one end of a plank of length L which lies at rest on a frictionless surface. The man walks to the other end of the plank. If the mass of the plank is $\frac{M}{3}$, the distance that the man moves relative to the ground is

The radii of two planets are respectively $R_1$ and $R_2$ and their densities are respectively ${\rho }_{1}and{\rho }_2$. The ratio of the accelerations due to gravity at their surface is

In the figure shown, find the direction of friction on block A.

If the length of second's pendulum is decreased by $2\%$, how many seconds it will loose per day ?

The equation of a stationary and a travelling waves are $y_1=a\sin kx\cos \omega t$ and $y_2=a\sin (\omega t-kx)$ respectively. The phase difference between two points, $x_1=\frac{\pi }{3k}$ and $x_2=\frac{3\pi }{2k}$ is ${\varphi }_1$ in the standing wave $y_1$ and is ${\varphi }_2$ in travelling wave $y_2$ then the ratio $\frac{{\varphi }_1}{{\varphi }_2}$ is

An engine generates a power of $75\text{kW}$ having efficiency of $80\%$ and the car moves with a constant velocity of $20\text{m/s}$. Find the force generated by the engine. (Assume the engine applies a constant force on the car)

A vessel contains oil (density $0.8{\text{g/cm}}^2$) over mercury (density $13.6{\text{g/cm}}^3$). A homogeneous sphere floats with half its volume immersed in mercury and the other half in oil. The density of the material of the sphere in ${\text{g/cm}}^3$ is

A ball of mass $m$ is suspended by a massless string of length $l=10\text{m}$ from a fixed point as shown in the figure below. A ball of mass $2m$ strikes at an angle $\theta ={45}^{\circ }$ from the horizontal and sticks to it. If the system deflects by angle $\varphi =\frac{\pi }{2}$, then the velocity of the system just after the collision and the velocity of $2m$ particle before collision are, respectively $\left(\text{in m/s}\right)$

The approximate depth of an ocean is $3500\text{m}$. The compressibility of water is $50\times {10}^{-11}{\text{Pa}}^{-1}$ and density of water is ${10}^3{\text{kg/m}}^3$. Find the fractional compression of water that will be obtained at the bottom of the ocean. (Take $g=10{\text{m/s}}^2$)

The figure shows at time $t=0\text{sec}$, a rectangular and triangular pulse on a uniform wire. They are approaching each other. The pulse speeds are $0.5\text{cm/s}$ each. The resultant pulse at $t=2\text{sec}$ is

The graph of displacement v/s time is shown in Fig



Its corresponding velocity-time graph will be

In the figure shown, what is the value of mass ‘$m$’ such that block $A$ slides up with a constant velocity. Take $g=10{\text{m/s}}^2$, Coefficient of friction between inclined plane and block $A$ is $0.5$.

A block is released from rest from a height $h=5\text{m}$. After travelling through the smooth curved surface, it moves on the rough horizontal surface through a length $l=8\text{m}$ and climbs onto the other smooth curved surface through a height $h^{{}^{\prime }}$. If coefficient of friction of the rough surface is $\mu =0.5$, find $h^{{}^{\prime }}$.

A block $B$ is pushed momentarily along the horizontal surface with an initial velocity $v$. If $\mu$ is the coefficient of sliding friction between $B$ and the surface, block $B$ will come to rest after a time

A system of solid discs, each of mass $M$ and radius $R$ is shown in figure.





Find the MOI about an axis passing through point $\text{O}$ and perpendicular to the plane.

Two bodies rotating with the same angular momentum have M.I. $I_{1}and{I}_2$ respectively such as $I_1>I_2$ respectively, then

A rectangular block with mass $2\text{kg}$ is sliding down an incline plane at angle of ${30}^{\circ }$. The block is separated from the plane by a lubrication oil with dynamic viscoscity $0.25{\text{Nsm}}^{-2}$ having spacing of $1\text{mm}$. The base of the block has an area of $20\text{cm}\times 50\text{cm}$. Determine the terminal velocity (when acceleration is zero) of the block as it slides down the plane

The equation of projectile is $y=8x-\frac{5x^2}{2}$. The horizontal range is-

The system starts from rest and block $A$ attains a velocity of $5\text{m/s}$ after it has moved $5\text{m}$ towards the right. Assuming the arrangement to be frictionless everywhere and pulley & strings to be light, find the value of the constant force $F$ applied on block $A$. (Take $g=10m/s^2$)

A square lamina of mass $m=4\text{kg}$ having sides of length $a=2\text{m}$. Find the moment of inertia of the lamina along one of its side.