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As shown in figure, a man of mass $M$ is standing on a platform and holding an inextensible massless string which passes over a system of ideal pulleys. Another body of mass $m$ is hanging as shown in figure. Find the force exerted by the platform on man if $M=50\text{kg},m=10\text{kg}$, Take $g=10{\text{m/s}}^2$.

The displacement x of a particle along a straight line at time t is given by $x=a_0+a_{1}t+a_2{t}^2$.The acceleration of the particle is

River is flowing at 3 Km/h in the west direction and You can swim at a maximum speed of 5 Km/h , then at what direction must you swim so that you end up swimming perpendicular to the bank?

A truck moving on a smooth horizontal surface with a uniform speed 'u' is carrying stone-dust. If a mass $\Delta m$ of the stone-dust `leaks down’ from the truck in the time $\Delta t$, the force needed to keep the truck moving at its uniform speed is

In the figure shown, if the block does not slide down the plane when a force of $\text{10 N}$ is applied perpendicular to the plane, find the coefficient of static friction between the plane and the block.(Consider limiting case of friction) Take $g={\text{10 m/s}}^2.$

Find the number of significant figures in the given data

$\begin{array}{cc}\text{P)}0.543& \text{U)}3\\ \text{Q)}4.330& \text{V) 4}\\ \text{R)}1.32\times {10}^{-2}& \text{W)}5\\ \text{S)}0.002& \text{X)}1\end{array}$

If $U_1,U_2$ represents "change in potential energy" of a body with mass $m$ moving from $A$ to $B$ along two different paths $1$, & $2$ (as shown in the figure) in the gravitational field. Then

Current $I_1$ and $I_2$ flow in the wires as shown in the figure. The field is zero at a distance $x$ to the right of O. Then,

In the figure shown, a block $A$ moving with velocity $20\text{m/s}$ on a horizontal surface collides with another identical block $B$, initially at rest. The coefficient of restitution is $0.5$. Neglect friction everywhere. The distance between the blocks, $5\text{sec}$ after the collision takes place is

Ratio of rotational kinetic energy to the total kinetic energy for diatomic gas molecule at room temperature is :

A body of mass m is taken to the bottom of a deep mine. Then

A body is falling under gravity from rest. When it loses gravitational potential energy by U, its speed is v. The mass of the body shall be

A piece of ice starting from rest slides down a rough ${45}^{\circ }$ incline in twice the time it takes to slide down a frictionless ${45}^{\circ }$ incline. What is the coefficient of friction between the ice and incline?

Linear mass density of a bar is $\rho =\frac{{\rho }_o}{L_o}L+{\rho }_o$ where ${\rho }_o$ is constant and $L_o$ is length of bar. Its centre of mass from one of the end is

(Assume the other dimensions of the bar to be unity)

A tube open at only one end is cut into two tubes of non-equal lengths. The piece open at both the ends has a fundamental frequency of $450\text{Hz}$ and the other piece has a fundamental frequency of $675\text{Hz}$. What is the first overtone frequency of the original tube?

As shown in the figure, pressure acting on the two limbs of a $U-$tube are $P\&P_0$ respectively. Then, identify the correct relation between $P\&P_0$:

The block $m_1$ is loaded on the block $m_2$. If there is no relative sliding between the blocks and the inclined plane is smooth, find the friction force between the blocks.

The ratio of the K.E. required to be given to the satellite to escape earth's gravitational field to the K.E. required to be given so that the satellite moves in a circular orbit just above earth atmosphere is

The interference pattern is obtained with two coherent light sources of intensity ratio $n$. In the interference pattern, the ratio $\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$ will be

A beaker containing a liquid of density $\rho$ moves up with an acceleration ${}^{\prime }{a}^{\prime }$. The pressure in the liquid at a depth $h$ below the free surface of the liquid is (consider atmospheric pressure to be $P_0$)

A plane electromagnetic wave of frequency $40\text{MHz}$ travels in free space in the X - direction. At some point and at some instant, the electric field $\overrightarrow{E}$ has its maximum value of $750\text{N/C}$ in Y -direction. The wavelength of the wave is-

The potential energy of a certain spring when stretched through a distance 'S' is 10 joule. The amount of work (in joule) that must be done on this spring to stretch it through an additional distance 'S' will be

The angular position of one of the blades of a fan is ${\theta }_1=\frac{\pi }{3}\text{rad}$, as shown in the figure. If the fan is switched on and off within a fraction of a second, the blade’s angular position changes to ${\theta }_2=\frac{\pi }{2}\text{rad}$. The angular displacement of all the blades is

A swimmer starts swimming in a river (width $=d$) to reach the other end of the river and river velocity is zero when he started the swimming. His velocity is $2.5m/s$ w.r.t. river and once he reaches half of the width of the river, then the river starts flowing with a velocity $2.5m/s$. What is the horizontal displacement of the swimmer from the initial position? ($d=400m$)

A long cylindrical rod of radius $R_1$ is placed co-axially inside a cylindrical tube of radius $R_2(>R_1)$ filled withviscous liquid and pulled along the axis with a constant velocity. In steady state the velocity gradient $\left(\frac{dv}{dr}\right)$ depends on the distance from the common axis 'r' as proportional to

For a fusion reaction between two light weight nuclei to occur, the Columbic repulsion between them is required to be overcome. This can be done by which of the following way/s ?

A particle starts from the point $(0m,8m)$ and moves with uniform velocity of $3\hat{i}\text{m/s}$. After 5 seconds, the angular velocity of the particle about the origin will be :

A straight wire carrying a current of 10 A is bent into a semi-circular arc of radius $\pi$ cm as shown in Fig. What is the magnitude and direction of the magnetic field at centre 0 of the arc?

If the incident ray is rotated by an angle of ${30}^{\circ }$ while the plane mirror is at rest, the reflected ray gets rotated by .

In case of projectile motion if two projectiles A and B are projected with same speed at angles ${15}^{\circ }$ and ${75}^{\circ }$ respectively to the horizontal then:

Water flows through two identical tubes A and B. A volume V₀ of water passes through the tube A and 2 V₀ through B in a given time. Which of the following may be correct?

Galileo writes that for angles of projection of a projectile at angles $(45+\theta )and(45-\theta )$, the horizontal ranges described by the projectile are in the ratio of $(if\theta \le 45)$

The motion of a particle along a straight line is described by the function $x=(2t-3{)}^2$ where $x$ is in meters and $t$ is in seconds. Find the velocity of the particle at the origin.

The time period of a body in $SHM$ is represented by $T_o=P^a{\rho }^b{S}^c$, where $P$ is pressure, $\rho$ is density and $S$ is surface tension. The value of $a,b$ and $c$ respectively for the relation to be dimensionally correct is

The energy spectrum of a blackbody exhibits a maximum around a wavelength ${\lambda }_0$. The temperature of the black body is now changed such that the energy is maximum around a wavelength $\frac{3{\lambda }_0}{4}$ The power radiated by the blackbody will now increase by a factor:

An object is displaced from point $A(0,0,0)m$ to a point $B(2,4,6)m$ under the influence of a force $\overrightarrow{F}=2x\hat{i}+3y^2\hat{j}+4z^3\hat{k}.$ The workdone by this force is

The given lens is broken into four parts rearranged as shown. If the initial focal length is $f$, then after rearrangement the equivalent focal length is

Find minimum value of $l$ so that truck can avoid the dead end, without toppling the block kept on it.

A political party has to start its procession in an area where wind is blowing at a speed of 41.4 km/h and party flags on the vehicles are fluttering along north-east direction. If the procession starts with a speed of 40 km/h towards north, find the direction of flags on the vehicles.

A projectile fired with initial velocity u at some angle $\theta$ has a range R. If the initial velocity be doubled at the same angle of projection, then the range will be

A particle of mass $2\text{kg}$ is moving in a circular path of constant radius $1\text{m}$ such that its centripetal acceleration $a_c$ is varying with time $t$ as $a_c=100t^2$. The power delivered to the particle by the force acting on it at $t=5\text{sec}$ is:

In the figure shown below, if the downward acceleration $a$ of $B$ is along the fixed incline, then find the accleration of $A$ if $A$ and $B$ remains in contact.

A point charge of magnitude q = 4.5 nC is moving with speed v = $3.6\times {10}^7$ m/s parallel to the x-axis along the line y = 3 m. Find the magnetic field at the origin produced by this charge when the charge is at the point x = -4 m, y = 3 m, as shown in the figure.

A pulley fixed to the ceiling carried a thread with bodies of masses $m_1$ and $m_2$ attached to its ends. The masses of the pulley and the thread are negligible and friction is absent. Find the acceleration of the center of mass of this system.

A body of mass $1\text{kg}$ is projected with velocity $50\text{m/s}$ at an angle of ${30}^{\circ }$ with the horizontal. At the highest point of its path, a force $10\text{N}$ starts acting on body for $5s$ vertically upwards besides gravitational force. What is horizontal range of the body ? ($g=10{\text{m/s}}^2$):

A particle of mass m was transferred from the centre of the base of a uniform hemisphere of mass M and radius R to infinity. How much work would be performed in this process by the gravitational force exerted on the particle by the hemisphere?

Find the density of a metallic body which floats at the interface of mercury of specific gravity $13.6$ and water such that $31.75\%$ of its volume is submerged in mercury and $68.25\%$ in water.