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A string is passing through a ring of mass $m$ as shown. The ring rotates in a horizontal plane of radius $r$ with a constant angular velocity. What should be the maximum speed of rotation so that the string does not get distorted?

If $se{c}^2\theta =\frac{4xy}{(x+y{)}^2}$ is true, then which of the following is true? $(x\ne -y)$

A block of mass 'm' and a pan of equal mass are connected by a string going over a smooth light pulley as shown in figure. Initially the system is at rest when a particle of mass m falls on the pan and sticks to it. If the particle strikes the pan with a speed 'v' find the speed with which the system moves just after the collision.

Moment of the force, $\overrightarrow{F}=(4\hat{i}+5\hat{j}-6\hat{k})\text{N}$ acting at point $P(2,0,-3)\text{m}$ about the point $Q(2,-2,-2)\text{m}$ is given by :

Find the magnetic field due to a square frame of side a carrying a current as shown at 0.

If a cycle wheel of radius 4 m completes one revolution in two seconds. Then acceleration of a point on the cycle wheel will be

Calculate the velocity of the image with respect to a stationary observer in the following situation (Object moving at an angle with mirror and mirror moving at an angle with object).

Two blocks are connected with a string and the system is connected to another string which is fixed at the centre of a circular table as shown in the figure. The table is rotating with an angular speed of $\omega$ about its centre. Find the tension in the string which is fixed at the centre.

The area of the parallelogram represented by the vectors $\overrightarrow{A}=2\hat{i}+3\hat{j}$ and $\overrightarrow{B}=\hat{i}+4\hat{j}$ as its sides is

In Young's double slit experiment using two identical slits, the intensity of the maximum at the centre of the screen is I. What will be the intensity at the centre of the screen if one of the slits is closed?

A milliammeter of range $10\text{mA}$ and resistance $9\Omega$ is joined in a circuit as shown. The meter gives full scale deflection for current $I$ when $A$ and $B$ are used as its terminals. i.e., current enters at $A$ and leaves at $B$ ($C$ is left isolated). The value of $I$ is

Add 3.1421 ,0.241 and 0.09 and give your answer with correct precision

The length of a steel wire when stretched by a force $F_1$ is $l_1$ and when stretched by a force $F_2$ is $l_2$.The unstretched length of the wire is

The molar heat capacity of a gas at constant volume is $5{\text{cal mol}}^{-1}{\text{K}}^{-1}$. Find the value of $\gamma$ for the gas if the gas constant, $R=2{\text{cal mol}}^{-1}{\text{K}}^{-1}$.

Block of $3\text{kg}$ is initially in equilibrium and is hanging by two identical springs $A$ and $B$ as shown in the figure. If spring $A$ is cut from lower point at $t=0$ then, find acceleration of block in $m{s}^{-2}$ at $t=0$.

A $1200\text{kg}$ automobile rounds a level curve of radius $200\text{m}$, on an unbanked road with a velocity of $72\text{km/h}$. What is the minimum coefficient of static friction between the tyres and road in order that the automobile may not skid? $(g=10{\text{m/s}}^2)$

A tuning fork produces $4$ beats per second with another tuning fork of frequency $266\text{Hz}$. The first one is now loaded with wax and the beat frequency is found to be $8$ per second. What is the original frequency of the first tuning fork ?

The magnifying power of a microscope with an objective of 5 mm focal length is 400. The length of its tube is 20 cm. Then the focal length of the eye-piece is [MP PMT 1991]

A body is moving up an inclined plane of angle $\theta$ with an initial kinetic energy E. The coefficient of friction between the plane and the body is $\mu$. The work done against friction before the body comes to rest is

A bob of mass $m=50\text{kg}$ is suspended from the ceiling of a trolley by a light inextensible string. If the trolley accelerates horizontally, the string makes an angle ${30}^o$ with the vertical. Find the acceleration of the trolley.

Two balls are dropped from different heights at different instants. Second ball is dropped $2\text{s}$ after the first ball. If both the balls reach the ground simultaneously, after $5\text{s}$ of dropping the first ball, then the difference between the initial heights of the two balls will be

$(\text{Take}g=9.8{\text{m/s}}^2)$

A tall cylinder is filled with viscous oil. A round pebble is dropped from the top with zero initial velocity. From the plot shown in the figure, indicate the one that represents the velocity $\left(v\right)$ of the pebble as a function of time $\left(t\right)$.

The work function of a metal is 4.2 eV, its threshold wavelength will be

A truck is moving with an acceleration of $2{\text{m/s}}^2$ due to a constant horizontal force $F$. Suddenly, a hail storm starts with a constant rate of mass $\mu =20\text{kg/s}$ getting deposited on the truck. Find the force needed to maintain the acceleration of the truck after $2$ seconds. Take $g=10{\text{m/s}}^2.$ Speed of the hailstorm is $2\text{m/s}$ (perpendicular to the velocity of the truck) and mass of the truck is initially $m_0=50\text{kg}$.

A rod of mass $m$ and length $2l$ hangs by two identical light threads tied to its ends. An insect of mass $\frac{3m}{8}$ lands on the rod gently at a distance $\frac{l}{3}$ from the centre of the rod as shown and sticks to it. As a result one of the thread breaks. The acceleration of the insect just after the thread breaks given that the insect remains at rest with respect to the rod.

An equal force of 100 N is applied upon two similar sharp nails at different angles as shown in figure ⇒

The ratio of P(I) : P(II) will be?

The average velocity of molecules of a gas of molecular mass $M$ at temperature $T$ is

A car is moving in direction $\overrightarrow{r}=-4\hat{i}+3\hat{j}$ with a speed of 10m/s. Write velocity vector of car in vector notation.

A small equilateral triangular loop of side $l=1cm$ is kept inside a large circular loop of radius r such that both are coplanar and concentric. If current in the larger loop is as shown in the graph, then find the current induced in the smaller loop during the time instants given. Take anticlock wise current as positive.





List - 1 gives time instants and list - 2 gives induced current in the smaller loop. Take $\frac{{\mu }_0\pi }{2rR}=1$ where R is resistance of small loop.

$\begin{array}{cccc}& \text{List -1}& & \text{List -2}\\ \left(I\right)& 0sec& \left(p\right)& 25\mu A\\ \left(II\right)& \frac{1}{3}sec& \left(Q\right)& 12.5\mu A\\ \left(III\right)& \frac{3}{2}sec& \left(R\right)& 0\mu A\\ \left(IV\right)& \frac{5}{3}sec& \left(S\right)& -6.25\mu A\\ & & \left(T\right)& -12.5\mu A\\ & & \left(U\right)& -25\mu A\end{array}$

A string of length $1.5\text{m}$ with both its ends clamped is vibrating in fundamental mode. Amplitude at the centre of string is $4\text{mm}$. What is the minimum distance between two points having amplitude $2\text{mm}$?

For a car moving on a straight line with initial speed $u$, the minimum distance over which it can be stopped is $s$. If it's initial speed becomes $nu$, what will be the minimum distance over which it can be stopped? Assume constant braking force.

A $200\text{Hz}$ sinusoidal wave is travelling in the positive $x$-direction along a string with a linear mass density of $4\times {10}^{-3}\text{kg/m}$ and a tension of $40\text{N}$. At time $t=0$, the point $x=0$ has maximum displacement in the positive $y$-direction. Next when this point has zero displacement, the slope of the string is $\frac{\pi }{20}$. Which of the following expressions respresent the displacement of string as a function of $x$ (in metre) and $t$ (in seconds).

A convex mirror has a focal length $f$. A real object, placed at a distance $f$ in front of it from the pole, produces an image at

What is the magnitude of minimum force $F$ necessary to push the block up against a wall, as shown in the figure. The coefficient of friction between the block and the wall is $0.5$, the weight of the block is $12\text{N}$.

A missile of mass 1 kg is moving with speed 10 m/s along the +ve x direction. Its target is a car weighing 100 kg, moving with 1 m/s also in the +ve x direction. The missile hits the target but doesn't burst. Find the combined velocity of car-missile system?

An electric dipole consisting of two opposite charges of $2\times {10}^{-6}C$ each, separated by a distance of 3 cm is placed in an electric field of $2\times {10}^{5}N/C$. The maximum torque on the dipole will be

Which of the following has more energy than X-ray:

An object is released from some height. Exactly after one second, another object is released from the same height. The distance between the two objects exactly after $2$ seconds of the release of second object will be

In one of the exercises to strengthen the wrist and fingers, a person squeezes and releases a soft rubber ball. Is the work done on the ball positive, negative or zero (i) during compression (ii) During expansion?

An object of m kg with speed of v m/s strikes a wall at an angle q and rebounds at the same speed and same angle. The magnitude of the change in momentum of the object will be

A particle executes SHM with a time period of $4\text{s}$. Find the time taken by the particle to go directly from its mean position to half of its amplitude.

A particle travels along the X-axis. Its velocity time graph is shown above.

At what time instant did the particle change its direction of motion?

Figure shows a cylindrical container containing oxygen gas and closed by a piston of mass $50\text{kg}$. Piston can slide smoothly in the cylinder. Its cross sectional area is $100{\text{cm}}^2$ and atmospheric pressure is ${10}^5\text{Pa}$. Some heat is supplied to the cylinder so that the piston is slowly displaced up by $20\text{cm}$.

Find the amount of heat supplied to the gas.

Thief A is in an apartment 10m high and is throwing a bag of valuables horizontally at 5 m/s. Thief B is initially standing right below the point of release. Find the velocity with which he should run to catch the bag in time.