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A log of wood of mass 120 Kg floats in water. The weight that can be put on the raft to make it just sink should be (density of wood=600Kg/m³)

When the speed of a car is $u$, the minimum distance over which it can be stopped is $s$. If the speed becomes $nu$, what will be the minimum distance over which it can be stopped during same time? Assume the acceleration is constant and same in both cases.

A solid cone has a height of $20\text{cm}$. Find the distance of center of mass of the cone from the center of the base of the cone.

A particle executing simple harmonic motion has angular frequency $6.28s^{-}1$ and amplitude $10cm$. Find $\left(x\right)$ the speed when the displacement is $6cm$, from the mean position, $\left(y\right)$ the speed at $t=\frac{1}{6}s$ assuming that the motion starts from rest at $t=0$.

(i) $25.1cm{s}^{-1}$

(ii) $54.4cm{s}^{-1}$

(iii) $50.2cm{s}^{-1}$

(iv) $27.2cm{s}^{-1}$

The figure shows a $U-$ tube of uniform cross-sectional area $A$ having a liquid of mass $M$ and density $\rho$. If the liquid in the column is slightly displaced, then it oscillates with a time period of :

[$g=$ acceleration due to gravity]

Torque is nothing but___.

Two charged spheres A and B charged with the charges of + 10 and +20 coulombs respectively and Separated by a distance of 80 cm. The electric field at a point on the line joining the centres of the two spheres will be zero at a distance from sphere A

If a photon has velocity c and frequency v, then which of following represents its wavelength

A body of mass $2\text{kg}$ has an initial velocity of $3m/s$ along $OE$ and it is subjected to a force of $4\text{N}$ in a direction perpendicular to $OE$. The magnitude of displacement of the body from $O$ after 4 seconds will be

A bob of a simple pendulum of mass $m$ is projected horizontally with some speed from its lowest point such that the string will slack at a position where it makes an angle ${60}^{\circ }$ with the upward vertical direction. Find the tension in the string at the lowest position.

The volume of containers A and B, connected by a tube and a closed valve are $V$ and $4V$ respectively. Both the containers A and B have the same ideal gas, at pressures $5.0\times {10}^5\text{Pa}$ and $1.0\times {10}^5\text{Pa}$ and temperatures $300\text{K}$ and $400\text{K}$ respectively. The valve is opened to allow the pressure to equalise, but the temperature of each container is kept constant at its initial value. Find the common pressure in the containers.

A spherical ball of mass $4m$, density $\sigma$ and radius $r$ is attached to a pulley-mass system as shown in the figure. The ball is released in a beaker with a liquid of coefficient of viscosity $\eta$ and density $\rho (<\frac{\sigma }{2})$. If the length of the liquid column in the beaker is sufficiently long, the terminal velocity attained by the ball is given by: (Assume all pulleys to be massless and strings to be massless and inextensible):

Figure shows a smooth vertical circular track of radius $R$. A block slides along the surface $AB$ when it is given a velocity equal to $\sqrt{6gR}$ at point $A$. The ratio of the force exerted by the track on the block at point $A$ to that at point $B$ is

A simple pendulum $50\text{cm}$ long is suspended from the roof of a cart accelerating in the horizontal direction with constant acceleration of $\sqrt{3}g{\text{m/s}}^2$. The period of small oscillations of the pendulum about its equilibrium position is -

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

Light of wavelength $\lambda$ falls on a metal having work function $\frac{hc}{{\lambda }_0}$. Photoelectric effect will take place only if

A body is moving with velocity $30\text{m/s}$. A variable force $a\text{N}$ is applied at $t=0\text{s}$ on the body such that it increases it's velocity to $90\text{m/s}$ at $t=4\text{s}$ in same direction. The variation of force with time is shown in graph.



If mass of body is $10kg$, find the value of $a$ (shown in graph)

Once a zero potential energy reference state is chosen, the changes in potential energy for a given initial and final states

Three particles of masses $50\text{g},100\text{g}$ and $150\text{g}$ are placed at the vertices of an equilateral triangle of side $2\text{m}$ as shown in the figure. The $(x,y)$ co-ordinates of the centre of mass (in $\text{m})$ will be

If force, F =$\frac{x\left(\mathrm{volume}\right)}{\left(\mathrm{speed}\right)\left(\mathrm{energy}\right)}$+ c , is a correct equation. Find the dimensions of x.

A particle is in linear simple harmonic motion between two points $P$ and $Q$ represented by $x=A\text{sin}(\omega t+\varphi )$. Find the initial phase of the particle when it starts from its extreme position.

A point mass m is attached at the end of a massless rod of length L. Other end of rod is hinged to frictionless hinge. Sudden velocity $V=\sqrt{\frac{9gL}{2}}$ is provided to the point mass as shown in figure. So hinge reaction when this rod rotates by ${180}^0$ is

The currect through a wire depends on time as,

$i=(10+4t)$

Here, $i$ is in ampere and $t$ in seconds. Find the charge crossed through a section in time interval between $t=0$ to $t=10\text{s}$.

A block of mass $10\text{kg}$ is kept on a rough inclined plane of inclination ${45}^{\circ }$. A force of $3\text{N}$ is applied on the block down the incline. The coefficient of static friction between the plane and the block is $0.6$. What should be the minimum value of force $F$, such that the block does not move downward? (Take $g=10{\text{ms}}^{-2}$)

The body started with velocity of $8\text{m/s}$ and moving with an acceleration of $1{\text{m/s}}^2$. The distance travelled by the body in $6^{\text{th}}$ second is

A uniform disc of mass m and radius R is projected horizontally with velocity $v_0$ on a rough horizontal floor, so that it starts off with a purely sliding motion at t=0. After $t_0$ seconds, it acquires a purely rolling motion as shown in figure.

Consider a gas with density $\rho$ and $\overline{c}$ as the root mean square velocity of its molecules contained in a volume. If the system moves as whole with velocity v, then the pressure exerted by the gas is

A light string passing over a smooth light pulley connects two blocks of masses $m_1$ and $m_2$ (vertically). If the acceleration of the system is g/8 then the ratio of the masses is

The density of water at the surface of the ocean is $\rho$. If the bulk modulus of water is B, what is the density of ocean water at a depth where the pressure is $n{P}_0$, where $P_0$ is the atmospheric pressure?

Two vectors have magnitudes 3 unit and 4 unit respectively. What should be the angle between them if the magnitude of the resultant is 5 unit

Again, the engine does 41,000 J of useful work. If it sinks 9,000 J of energy in the form of heat to its cold reservoir, what is the efficiency of this engine?

The absolute zero is the temperature at which :

A vessel A has volume V and a vessel B has volume 2V. Both contain some water which has a constant volume. The pressure in the space above water is $p_a$ for vessel A and $p_b$ for vessel B.

A rail-road car is moving towards the right with acceleration $a$ and a dog of mass $m$ is chasing a man $\left(A\right)$ with an acceleration $\frac{a}{3}$ relative to the car and the man $\left(A\right)$ is accelerating towards left with an acceleration of magnitude $\frac{a}{3}$ with respect to the car as shown in the figure. If an observer $B$ on the ground is observing both the dog and man $\left(A\right)$, then the net force experienced by the dog as seen by observer $B$ is

The radius vector of a point A relative to the origin varies as $\overrightarrow{r}=at\hat{i}+b{t}^2\hat{j}$ where a and b are positive constants. Find the equation of trajectory.

A spring block system is compressed $x\text{m}$ from the mean position and released. Find out the total work done by the spring force when it reaches the other extreme point (fully stretched). Neglect friction.

In which time duration is the (magnitude of velocity) speed of the particle increasing?

Two equal forces are acting at a point with an angle of ${60}^{\circ }$between them. If the resultat force is equal to $40\sqrt{3}\text{N}$, the magnitude of each force is

Column I of the following table gives a list of bodies and column II gives their moments of inertia about the axis of symmetry passing through their centre of mass. Assume mass of each of the bodies is $M$. Then identify the correct matching.

A horse is harnessed to a cart. If the horse tries to pull the cart, the horse must exert a force on the cart. By Newton's third law the cart must then exert an equal and opposite force on the horse. Since the two forces are equal and opposite, they must add to zero, so Newton's second law tells us that the acceleration of the system must be zero and therefore no matter how hard the horse pulls, it can never move the cart.

Find the force of interaction between $2kg$ and $1kg$ bodies for the figure shown. (Take $g=10m/s^2$)

A metallic cylindrical vessel whose inner and outer radii are $r_1$ and $r_2$ is filled with ice at $0^{\circ }\text{C}.$ The mass of the ice in the cylinder is $m$. Circular surfaces of the cylinder are sealed completely with adiabatic walls. The vessel is kept in air and the temperature of the air is ${50}^{\circ }\text{C}.$ How long will it take for the ice to melt completely?

[Thermal conductivity of the cylinder is $K$ and its length is $l$. Latent heat of fusion of ice is $L$].

A small block oscillates back and forth on a smooth concave surface of radius R. The time period of small oscillation is

A particle with charge q and mass m is projected with kinetic energy K into the region between two plates of a uniform magnetic field B as shown below.



If the particle is to miss collision with the opposite plate, the maximum value of B is

Velocity of a car as a function of time is given by $v\left(t\right)=(4t^3-e^t+\sin t)$ in $\text{m/s}$. The acceleration in $\left({\text{m/s}}^2\right)$ of the car as a function of time $t$ is

A moving railway compartment has a spring of constant k fixed to its front wall. A boy stretches this spring by distance x and in the mean time the compartment moves by a distance s. The work done by boy w.r.t. earth is

Two Cars $A$ and $B$ are travelling with constant velocities of $10\text{m/s}$ and $20\text{m/s}$ respectively along straight lines making an angle of $60°$ with each other. The magnitude of relative velocity of $A$ with respect to $B$ is