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A cone of radius R and height H, is hanging inside a liquid of density $\rho$ by means of a string as shown in the figure . The force, due to the liquid acting on the slant surface of the cone is (Neglect atmosphere pressure)

A string fixed at both ends is vibrating in seventh harmonic. How many nodes and antinodes are set up in it respectively?

The fundamental frequency of a closed organ pipe of length 20 cm is equal to the second overtone of an organ pipe open at both the ends. The length of organ pipe open at both the ends is :

Light of wavelength $2475A^{\circ }$ is incident on Barium. Photoelectrons emitted describe a circle of radius 100 cm by a magnetic field of flux density $\frac{1}{\sqrt{17}}\times {10}^{-5}$ Tesla. Work function of the barium is (Given $\frac{e}{m}=1.7\times {10}^{11})$

The sun emits maximum radiation of $0.52$ micrometer. Assuming the sun to be a black body, the surface temperature of the sun would be (Wien's constant $=2.898\times {10}^{-3}mK$)

A cone of radius R and height H, is hanging inside a liquid of density by means of a string as shown in the figure . The force, due to the liquid acting on the slant surface of the cone is (Neglect atmosphere pressure)

A mass m is undergoing SHM in vertical direction about the mean position with amplitude A and angular velocity $\omega$ . At a distance y from the mean position, the mass detaches from the spring. Assume that the spring contracts and does not obstruct the motion of m. Find the distance $y_0$ (measured from the mean position) such that the height attained by the block is maximum $(A{\omega }^2>g)$

When a force is applied at one end of an elastic wire, it produces a strain E in the wire. If Y is the Young's modulus of the material of the wire, the amount of energy stored per unit volume of the wire is given by

Find the second derivative of the following function: $y=e^{3x+2}$

Two rigid boxes containing different ideal gases are placed on a table. Box $A$ contains one mole of nitrogen at temperature $T_0$, while box $B$ contains one mole of helium at temperature $\left(\frac{7}{3}\right)T_0$. The boxes are then put into thermal contact with each other, and heat flows between them until the gases reach a common final temperature (Ignore the heat capacity of boxes). Then, the final temperature of the gases, $T_f$ in terms of $T_0$ is -

A ball of mass $4\text{kg}$ moving with velocity $4\text{m/s}$ collides with another identical ball at rest. The kinetic energy of the balls after collision is $\frac{7}{8}$ times of the original. Find the coefficient of restituton $\left(e\right)$.

Four particles of masses $m_1=2m,m_2=4m,m_3=m$ and $m_4$ respectively are placed at the four corners of a square. What should be the value of mass $m_4$ so that the centre of mass of the system of particles lies at the centre of square?

A uniform rod of length $l$ and mass $m$ is free to rotate in a vertical plane about $A$. The rod initially in horizontal position is released. The initial angular acceleration of the rod is (Moment of inertia of rod about $A$ is $\frac{m{l}^2}{3}$)

Why does lighting strikes only at sharp edges like leaves of trees??

End $A$ of a rod $AB$ is being pulled on the floor with a constant velocity $v_0$ as shown. Taking the length of the rod as $l$, at an instant when the rod makes an angle ${37}^{\circ }$ with the horizontal, calculate

The angular velocity of the rod

Two satellites $S_1$ and $S_2$ revolve round a planet in coplanar circular orbits in the same sense. Their periods of revolution are 1 hour and 8 hours respectively. The radius of the orbit of $S_1={10}^4$km When $S_2$ is closest to $S_1$ find speed of $S_2$ relative to $S_2$ and the angular speed of $S_1$ actually observed by an astronaut at $S_1$.

The elements of sets A , B and C are all colours as follows:

A = { Red , Orange , Yellow , Brown } ,

B = { Blue , Yellow , Green , Brown } ,

C = { Black , Brown , White , Grey }

Then, A U B U C is

An air bubble in a glass slab $(\mu =1.5)$ is 6 cm deep when viewed from one face and 4 cm deep when viewed from the opposite face. The thickness of the glass plate is3

Free body diagram of a car taking turn on the plane road is

An uniform rod of density $\rho$ is placed in a wide tank containing a liquid of density ${\rho }_0({\rho }_0>\rho )$. The depth of the liquid in the tank is half the length of the rod. The rod is in equilibrium, with its lower end resting on the bottom of the tank. In this position, the rod makes an angle $\theta$ with the horizontal. Then, identify the correct relation :

A particle with a specific charge $s$ is fired with a speed $v$ towards a wall at a distance $d$ perpendicular to the wall. What minimum magnetic field must exist in this region for the particle to not to hit the wall?

Statement 1: In order to make a body rotate, the net external force applied can be zero.

Statement 2: A body can even rotate in absence of any external force.

If a body moving in a circular path maintains constant speed of $10\text{m/s}$, then which of the following graphs correctly describes the relation between acceleration and radius?

A motorcyclist starts from the bottom of a slope of angle ${45}^{\circ }$ to cross the valley $PR$ as shown in the figure. The width of the valley is $90\text{m}$ and length of the slope is $80\sqrt{2}\text{m}$. The minimum velocity at point $O$ required to clear the valley will be

A body cools from ${60}^{\circ }C$ to ${50}^{\circ }C$ in 10 minutes when kept in air at ${30}^{\circ }C$.
In the next 10 minutes, its temperature will be___.

Find the radius of gyration of a uniform circular disc of radius $R=1\text{m}$ and mass $M=2\text{kg}$ about its axis passing through the edge and normal to the disc as shown in figure.

The adiabatic elasticity of a gas is equal to

A gun fires a bullet of mass 50 gm with a velocity of 30 metre per sec. Because of this the gun is pushed back with a velocity of 1 metre per sec . The mass of the gun is

A ring of radius $1\text{m}$ is performing combined translational and rotational motion on a frictionless horizontal surface with an angular velocity of $3\text{rad/s}$ as shown in figure. Find out velocity of the point $A$ in $\text{m/s}$, if the speed of the lowest point $V_P$ is $1\text{m/s}$.

If $1+\cos \alpha +{\cos }^2\alpha +....\infty =2-\sqrt{2},$ then $\alpha ,(0<\alpha <\pi )$ is

Suppose in an imaginary world the angular momentum is quantized to be even intergral multiples of $\frac{h}{2\pi }$. What is the longest possible wavelength emitted by hydrogen atoms in visible range in such a world according to Bohr's model ?

A wire has resistance $12\Omega$. It is bent in the form of a circle. The effective resistance between the two points on any diameter is equal to

A hypothetical magnetic field existing in a region is given by $\underset{B}{\to }=B_0\underset{e_r}{\to }$ where $\underset{e_r}{\to }$ denotes the unit vector along the radial direction. A circular loop of radius a, carrying a current i, is placed with its plane parallel to the x - y plane and the centre at (0, 0, d). Find the magnitude of the magnetic force acting on the loop.

A free atom of iron emits $K_{\alpha }$ X-rays of energy 6.4 keV. Calculate the recoil kinetic energy of the atom. Mass of an iron atom = 9.3 $\times {10}^{-26}$ kg.

Water flows through a horizontal tube as shown in figure. If the difference of heights of water column in the vertical tubes is 2 cm, and the areas of cross section at A and B are 4 cm² and 2 cm² respectively, find the rate of flow of water across any section. Given approx.

An air bubble of diameter 2 mm rises steadily through a solution of density 1750 kg m^-3 at the rate of 0.35 cm s^-1. Calculate the coefficient of viscosity of the solution. The density of air is negligible.

If $\overrightarrow{a}\text{and}\overrightarrow{b}$ are the position vectors of $\overrightarrow{A}\text{and}\overrightarrow{B}$ respectively, then find the position vector of point $\overrightarrow{C}in\overrightarrow{BA}$ produced such that $\overrightarrow{BC}=1.5\overrightarrow{BA}$.

A block of mass M slides along the sides of a bowl as shown in the figure. The walls of the bowl are frictionless and the base has coefficient of friction 0.2. If the block is released from the top of the side, which is 1.5 m high, where will the block come to rest ? Given that the length of the base is 15 m

An iron ball and a wooden ball of the same radius are released from the same height in vacuum. They take the same time to reach the ground. The reason for this is

From a water fall, water is falling at the rate of 100 kg/s on the blades of turbine. If the height of the fall is 100m then the power delivered to the turbine is approximately equal to