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Two blocks of mass $m_1$ and $m_2$ are connected by a weightless rod on a plane having inclination of ${37}^{\circ }$. The coefficients of dynamic friction of $m_1$ and $m_2$ with the inclined plane are $\mu =0.25$. Then the common acceleration of the two blocks and the tension in the rod are

The electric potential varies in space according to the relation $V=3x+4y.$ A particle of mass $0.1kg$ starts from rest from point $(2,3.2)$ under the influence of this field. The charge on the particle is $+1\mu C.$ Assume V and $(x,y)$ are in S.I. units

The component of electric field in the x - direction $\left(E_x\right)$ is

A body of mass $5kg$ is moving along the x-axis with a velocity $2m{s}^{-1}$. Another body of mass $10kg$ is moving along the y-axis with a velocity $\sqrt{3}m{s}^{-1}$. They collide at the origin and stick together. The final velocity of the combined mass is

Two pendula have time periods T and $\frac{5T}{4}$. They start SHM at the same time from the mean position. The phase difference between them after the pendulum with larger period completes one oscillation is

A bomb is dropped from an aeroplane moving horizontally at constant speed. When air resistance is taken into consideration, the bomb

Absolute scale of temperature is reproduced in the laboratory by making use of a

A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed $\omega$, as shown int he figure. At time t = 0, a small insect starts from O and moves with constant speed v with respect to the rod towards the other end. It reaches the end of the rod at t = T and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque $\left(\overrightarrow{\tau }\right)$ on the system about O, as a function of time is best represented by which plot?

A balloon filled with hydrogen has a volume of 1 $m^3$ and its mass is 1 kg.The volume of the block of a very light material which it can just lift is

[Density of material of block is 91.3 kg/$m^3$ and that of air is 1.3 kg/$m^3$]

A tuning fork $T$ produces 4 beats per second with a source ${}^{\prime }{S}^{\prime }$. On loading wax on $T$, it gives $5$ beats per second with the same source. When the source ${}^{\prime }{S}^{\prime }$ moves towards a fixed wall with a speed $5\text{m/s}$ then again $T$ gives $5$ beats per second with echo of the source. If speed of sound is $340\text{m/s}$, then frequency of the tuning fork, before loading wax, is

Two metal wires of identical dimensions are connected in series. If ${\delta }_1$ and ${\delta }_2$ are the conductivities of the metal wires respectively, the effective conductivity of the combination is

A rod of length L is rotating with an angular velocity $\omega$ in a uniform magnetic field $B_0$ as shown in figure. Potential difference between the ends of rod A and B is

(Take $OB=\frac{L}{4}$)

A bomb of mass $12\text{kg}$ is dropped by a fighter plane moving horizontally with a speed of $100{\text{ms}}^{-1}$ from a height of $1\text{km}$ from the ground. The bomb exploded after $10\text{s}$ into two pieces of masses in the ratio $1:5$. If the small part started moving horizontally with a speed of $600{\text{ms}}^{-1},$ the speed of bigger part will be $(g=10{\text{ms}}^{-2})$

A thin uniform rod of mass $M$ and length $L$ has its moment of inertia $I_{rod}$ about its perpendicular bisector. This rod is bend in the form of a circular shape of radius $r$. Now, circle has its moment of inertia $I_{\text{circle}}$ about its centre and perpendicular to its plane. Find the ratio of $I_{rod}:I_{\text{circle}}$.

A spring of force constant $K$ is cut into lengths of ratio 1 : 2 : 3. They are connected in series and the new force constant is $K^{{}^{\prime }}$. Then, they are connected in parallel and the force constant is $K^{\prime \prime }$. Then, $\frac{K^{{}^{\prime }}}{K^{\prime \prime }}$ is

Determine the rms value of a semi-circular current wave which has a maximum value of a Ampere. The y axis denotes current in Ampere and x axis denotes time.

A block of mass $5\text{kg}$ is moving in x-direction with a constant speed of $10\text{m/sec}$. It is subjected to a retarding force $F=-0.2x\text{joule /metre}$ during its travel from $x=20\text{metre}$ to $x=30\text{metre}$. Its final kinetic energy will be

In the manometer as shown, $2$ immiscible fluids - mercury $(\rho =13600{\text{kg/m}}^3)$ and water $(\rho =1000{\text{kg/m}}^3)$ are used as manometric fluids. The water end is exposed to the atmosphere $\left(100\text{kPa}\right)$ and the mercury end is exposed to a gas. At this position, the interface between the fluids is at the bottom most point of the manometer. The value of $h$ is found to be $9.45\text{m}$. The height of the mercury column is given to be $75\text{cm}$. Find the gauge pressure of the gas. $(g=9.8{\text{m/s}}^2)$

Find the position of the center of mass of a hollow hemisphere of radius $R=10\text{cm}$. Assume center $\text{O}$ as origin.

Which of these are unit vectors

Two points $P$ and $Q$ are diametrically opposite on a disc of radius $R$. A particle at point $P$ has linear velocity $v$ as shown in figure. Find the angular speed of particle wrt an axis passing through $Q$ and perpendicular to plane of the disc.

A particle's position in the x-y plane varies as $x\left(t\right)=3t^3+4t^2,y\left(t\right)=16t^4+2$



Find the acceleration vector of the particle after 1 sec of start.

[MP PMT 1994, 95; DCE 1999, 2001; AIIMS 2001]

A helium nucleus makes a full rotaion in a circle of radius $0.8\text{m}$ in two seconds. The value of the magnetic field $B$ at the centre of the circle will be

Gauss law for magnetic field fetches a flux identically equal to zero because

For a convex mirror :

Find the magnitude and direction of $\overrightarrow{A}\times \overrightarrow{B}$ and $\overrightarrow{B}\times \overrightarrow{A}$ respectively, if $\overrightarrow{A}$ and $\overrightarrow{B}$ lies in the $x-y$ plane as shown in the figure. Given $|\overrightarrow{A}|=3$ and $|\overrightarrow{B}|=2$

The graph between the displacement $x$ and time $t$ for a particle moving in a straight line is shown in figure. During the interval $OA,AB,BC$ and $CD$, the acceleration of the particle is

If the displacement $x$ and velocity $v$ of a particle executing simple harmonic motion are related through the expression $4v^2=25-x^2$, then its time period is

A given volume of air at ${37}^{\circ }\text{C}$ exerts a pressure of $870\text{mm}$ of mercury. The temperature at which the pressure becomes $1740\text{mm}$ of $Hg$ is

Two balls each of mass $1\text{kg}$ are connected to both ends of a uniform circular rod of mass $5\text{kg}$ and length $2\text{m}$. Find the moment of inertia of the system about an axis passing through the middle of the rod and perpendicular to the length of the rod.

A metre stick swinging in a vertical plane about a fixed horizontal axis passing through its one end undergoes small oscillations of frequency $f_0$. If the bottom half of the stick is cut off and glued to the upper half, then the new frequency for small oscillations is

A closed organ pipe, of length $1.5\text{m}$ and filled with a gas, is resonated in its fundamental mode with the help of a tuning fork. Another open pipe of same length but filled with air resonates with the same fork in its fundamental mode. The room temperature is ${30}^{\circ }\text{C}$. If the speed of sound in air at ${30}^{\circ }\text{C}$ is $340\text{m/s}$, then the speed of sound in the first gas at ${30}^{\circ }\text{C}$ will be

The belt of an electrostatic generator is $50cm$ wide and travels $30cm/s$. The belt carries charge into the sphere at a rate corresponding to ${10}^{-4}A$. What is the surface density of charge on the belt?

In equation $y=x^2{\cos }^{2}2\pi \frac{\beta \gamma }{\alpha }$, the units of $x,\alpha ,\beta$ are ${\text{m, s}}^{-1}$ and $(m{s}^{-1}{)}^{-1}$ respectively. The units of $y$ and $\gamma$ are

A boy throws a water-filled balloon at an angle of ${53}^{\circ }$ with a speed of $10m/s$. A car is advancing towards the boy at a constant speed of $5m/s$. If the balloon is to hit the car, how far away should the car be when the balloon is thrown? $(g=10m{s}^{-2})$

A block of mass $m$ slides without friction down a fixed inclined board of inclination $\alpha$ with the horizontal. After leaving the inclined, the block falls on a cart of mass $M$ filled with sand as shown.

Initial height of the block above the level of the cart is $h$ as shown. The velocity of cart when block drops on it will be



(Assume that block and cart move together, when block drops on to the cart)

How many forces would be acting on $M$ when its $FBD$ is considered?

Two balls marked 1 and 2 of the same mass m and a third ball marked 3 of mass M are arranged over a smooth horizontal surface as shown in the figure. Ball 1 moves with a velocity $v_1$ towards balls 2 and 3. All collisions are assumed to be elastic. If M < m, the number of collisions between the balls will be

A wire 4 m long and 0.3 mm diameter is stretched by a force of 100N. If the extension in the wire is 0.3 mm, calculate the potential energy stored in the wire.

A reversible engine converts one-sixth of the heat input into work. When the temperature of the sink is reduced by ${62}^{\circ }C$, the efficiency of the engine is doubled. The temperatures of the source and sink are

A concave mirror has a focal length of $20\text{cm}$. The distance between two possible positions of the object for which the image size is double of the object size is

Two uniform identical rods each of mass $M$ and length $L$ are joined to form a cross as shown in the figure. Find the moment of inertia of the cross about a bisector in the plane of rods as shown by dotted line in the figure.

Three particles of masses $2\text{kg}$, $3\text{kg}$ and $5\text{kg}$ are placed at the three vertices $A,B,C$ of a right angled triangle respectively. Co-ordinates of the particles are $A(0,3),B(0,0)$ and $C(3,0)$. Find the position of centre of mass of the particles.

A ship $A$ is moving eastwards with a speed of $20\text{km/h}$ and a ship $B$ ,$50\text{km}$ south of $A$ is moving northwards with a speed of $20\text{km/h}$. The time after which the distance between ships become shortest is

The density of the material of a cube is measured by measuring its mass and length of its side. If the maximum error in the measurement of mass and the length are $3\%$ and $2\%$ respectively, the maximum error in the measurement of density is

A thin semicircular conducting ring of radius R is falling with its plane vertical, in a horizontal magnetic induction B. At the position MNQ, the speed of the ring is V and the potential difference developed across the ring is

An object moves along the grid through the points A, B, C, D, E and F as shown in the figure. Find the magnitude of displacement covered by the moving object.