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A small block of mass $m$ slides along a smooth frictionless track as shown in the figure.

(i) If it starts from rest at $P$, what is the resultant force acting on it at $Q$?

(ii) At what height above the bottom of the loop should the block be released so that the force it exerts against the track at the top of the loop equals its weight?

A bullet of mass $50\text{g}$ leaves a rifle at an initial velocity of $2000\text{m/s}$ and strikes the earth at the same level with a velocity of $1000\text{m/s}$. The work done in overcoming the resistance of air will be

A small ball is rolled with speed $u$ from point $A$ along a smooth circular track as shown in figure. If $x=3R$, then determine the point of projection after passing through $C$, the highest point.

A ball is moving with a velocity of $8\text{m/s}$ towards a surface which is also moving in a downward direction with velocity of $2\text{m/s}$ as shown in figure. After collision with the surface, ball rebounds back. If the coefficient of restitution is $e=0.5$, find the rebound velocity of the ball. (Given:Velocity of surface will be same even after collision and all other external forces are absent.)

A metallic bob of mass ‘$m$’ attached to a string, is raised through a height of $50\text{cm}$ and released. At what distance along the vertical from the point of suspension should a nail be placed so that the bob just completes a circle with the nail as the centre?

Which of the following statements is/are correct regarding the refractive index ?

A small sphere of radius $r$ falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity, is proportional to

A geostationary satellite is orbiting the earth at a height of 6R above the surface of the earth; R being the radius of the earth. What will be the time period of another satellite at a height 2.5 R from the surface of the earth?

The number of photo-electrons emitted per second from a metal surface increases when

Centre of mass of three particles of masses $1\text{kg},2\text{kg}$ and $3\text{kg}$ lies at the point $(1,2,3)$ and centre of mass of another system of particles $3\text{kg}$ and $2\text{kg}$ lies at the point $(-1,3,-2)$. Where should we put a particle of mass $5\text{kg}$ so that the centre of mass of entire system lies at the centre of mass of $1^{st}$ system?

Unit vector perpendicular to $x-y$ plane is

A longitudinal wave is produced on a slinky toy. The wave travels at a speed of $30\text{cm/s}$ and the frequency of the wave is $20\text{Hz}$. What is the minimum separation between two consecutive compressions of the slinky toy ?

The potential energy (in joule) of a body of mass $2\text{kg}$ moving in the $x-y$ plane is given by $U=6x+8y$, where the position coordinates $x$ and $y$ are measured in metres. The body is at rest at point $(6\text{m},4\text{m})$ at time $t=0$. It will cross the $y-$ axis at time $t$ equal to

A semicircular portion of radius $r=10\text{cm}$ is cut from a uniform rectangular plate as shown in the figure. The distance of center of mass $\text{C}$ of the remaining plate, from point $\text{O}$ is (COM of semicircle is at distance $\frac{4r}{3\pi }$ from the base)

Two uniform rods of equal lengths but different masses are rigidly joined to form an $L$- shaped body, which is then pivoted about $\text{O}$ as shown in the figure. Find the net torque about point $\text{O}$. Given $M=m\sqrt{3}$

If the capillarity rise in two tubes containing same liquid and made of same material are $6.6\text{cm}$ and $2.2\text{cm}$, then the ratio of the radii of the tubes is

A litre of alcohol weighs

A train is travelling at the speed of 180 km/hr. In order to trap superman Lex Luthor has broken the tracks. Unware of Luthor's evil intensions, superman tries to stop the train by pushing it in opposite direction.Right at the moment when the train is 300 m away from the crack in the track, Superman is applying a deceleration of 5 m/$s^2$. Just after 2 sec a bullet laced with kryptonite hits superman which incapacitates him for a while. How long has super man got to remove the bullet and start to give deceleration so to protect the innocents? (Trains brake system has failed)

Two bodies A and B of equal masses are suspended from two separate springs of force constants $k_{1}and{k}_2$ respectively. If the two bodies oscillate such that their maximum velocities are equal, the ratio of the amplitudes of oscillation of A and B will be

A circular hole of diameter 0.15 cm is placed 2.1 m from a screen. On illuminating it by light of wavelength $5\times {10}^{-7}m$ the width of central maxima will be

If the coefficient of friction between $A$ and $B$ is $\mu$, the maximum acceleration of the wedge $A$ for which $B$ will remain at rest with respect to the wedge is:

One quarter sector is cut from a uniform circular disc of radius $R$. This sector has mass $M$. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

A $10\text{cm}$ long wire is placed horizontal on the surface of water and is gently pulled up with a force of $2\times {10}^{-2}\text{N}$ to keep the wire in equilibrium. The surface tension (in ${\text{Nm}}^{-1}$) of water is

Two identical capacitors are charged to different potentials, and then they are connected to each other in such a way that the Sum of charges on plates having positive polarity remains constant. Mark the correct statement:

Identify the incorrect statement regarding the reflection of light from a plane mirror when a light ray is incident obliquely. $\overrightarrow{R},\overrightarrow{I}$ and $\overrightarrow{N}$ represents the vectors along the reflected ray, incident ray and normal direction.

A block is attached to a spring having equilibrium position at $x=0$. The block oscillates between $x=-4\text{m}$ and $x=4\text{m}$. Find the change in potential energy stored in the spring when the block moves from $x=2$ to $x=-3$. Given the spring constant is $5\text{N/m}$.

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$, $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}-\hat{k}$ be three vectors. A vector $\overrightarrow{v}$ in the plane of $\overrightarrow{a}$ and $\overrightarrow{b}$, whose projection on $\overrightarrow{c}$ is $\frac{1}{\sqrt{3}}$, is given by

Given the magnitude of vector A to be 6 units. Find the component of A along the x-axis?

The volume of a gas at ${200}^{\circ }C$ is 200 ml. If the temperature is reduced to $-{20}^{\circ }C$ at constant pressure, its volume will be

Water rises in a capillary tube to a certain height such that the upward force due to surface tension is balanced by $75\times {10}^{-4}\text{N}$ force due to the weight of the liquid. If the surface tension of water is $6\times {10}^{-2}\text{N/m}$, the inner circumference of the capillary tube must be: (Take $\theta =0^{\circ })$

Two rods made of different materials are placed between massive walls (see fig.). The cross-section of the rods is A and their respective lengths are $l_1$and $l_2$. The rods are heated through $t^o$C. Find the force with which the rods act on each other if their coefficients of expansion are ${\alpha }_1$ and ${\alpha }_2$ and their Young's moduli are $E_1$ and $E_2$. Neglect the deformation of the walls

A pulley wheel of diameter $10\text{cm}$ has a $1\text{m}$ long cord wrapped around its periphery. The wheel is initially at rest. Assuming angular acceleration of wheel to be uniform and the time taken for the cord to unwind completely to be $18\text{s}$, find its tangential acceleration.

A soap bubble having radius $1mm$ is blown from a detergent solution having a surface tension of $2.5\times {10}^{-2}N/m$. The pressure inside the bubble is equal to the pressure at a point which is $Z_0$ below the free surface of water in a container. Find the value of $Z_0$ . Taking $g=10m/s^2$, density of water = ${10}^{3}kg/m^3$

Two particles A and B are moving in XY plane. Their positions vary with time $t$, according to relation

$x_A\left(t\right)=4t^2,x_B\left(t\right)=7$

$y_A\left(t\right)=3t,y_B\left(t\right)=3+4t^2$

Distance between these two particles at $t=1\text{s}$ is:-

In $v-t$ graph shown in fig. the displacement of the body in $5$ seconds will be, if it starts from origin.

A particle of mass $m$ is placed on top of a smooth hemispherical wedge of mass $4m$ and radius $R$ at rest. The hemispherical wedge is kept on a frictionless horizontal surface. The particle is given a gentle push. The angular velocity of the particle relative to the wedge, if wedge has velocity $v$ when particle has falls by an angle $\theta$ with respect to the wedge is

If the focal length of objective and eye lens are 1.2 cm and 3 cm respectively

and the object is put 1.25 cm away from the objective lens and the final

image is formed at infinity. The magnifying power of the microscope is

A man moving with velocity $5\text{m/s}$ along a straight line observes rain falling vertically at the rate of $10\text{m/s}$. Find the speed of rain with respect to ground, and angle$\left(\theta \right)$ made from vertical by the rain as seen from ground.

An elevator which is moving with an upward acceleration a. A coin is dropped from rest from the roof of the elevator, relative to you after what time the coin will strike he base of he elevator?

In the letter $E$ whose dimensions are given in the figure, origin is at the bottom left corner. Area of each rectangle is $12{\text{cm}}^2$ and area of the square is $4{\text{cm}}^2$. Assume that the weights are proportional to areas. Then, centre of gravity of the $E$ frame will be at:

The photoelectric threshold wavelength of a certain metal is $3000\stackrel{\circ }{A}$. If the radiation of $2000\stackrel{\circ }{A}$ is incident on the metal

The transition from the state n = 4 to n = 3 in a hydrogen-like atom results in ultra violet radiation. Infrared radiation will be obtained in the transition. (IIT-JEE-2001)

A liquid $X$ of density $3.36\text{g/cc}$ is poured in a $U-$ tube (containing mercury) in the right arm upto height $10\text{cm}$ above the mercury level. Another liquid $Y$ is poured in the left arm upto height $8\text{cm}$ as shown in figure. Upper levels of $X$ and $Y$ are the same. What is the density of liquid $Y$? [Take ${\rho }_{Hg}=13.6\text{g/cc}$]

Light of frequency v is incident on a certain photoelectric substance with threshold frequency $v_0$. The work function for the substance is

The resultant force of $5\text{N}$ and $10\text{N}$ cannot be