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The upper half of an inclined plane of inclination $\theta$ is perfectly smooth while the lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom if the coefficient of friction between the block and the lower half of the plane is given by

A capillary tube of radius $r$ is immersed in water and water rises in it to a height $h$. The mass of the water in the capillary is $5\text{g}$. Another capillary tube of radius $2r$ is immersed in water. The mass of water that will rise in this tube is :

A proton is moving along Z-axis in a magnetic field. The magnetic field is along X-axis. The proton will experience a force along

A
closely wound solenoid of 800 turns and area of cross section
2.5 × 10⁻⁴
m²
carries a current of 3.0 A. Explain the sense in which the solenoid
acts like a bar magnet. What is its associated magnetic moment?

Water is flowing through two horizontal pipes of different diameters which are connected together. The diameter of the two pipes are $3\text{cm}$ and $6\text{cm}$ respectively. If the speed of water in the narrower tube is $4\text{m/s}$, then the speed of water in the wider tube is

If $E_0$ and $B_0$ are the amplitudes of electric and magnetic field vectors in an electromagnetic wave, then the ratio of energy densities due to both fields is

Two sound waves of wavelength $1\text{m}$ and $1.01\text{m}$ in a gas produce $10$ beats in $3\text{s}$. The velocity of sound in the gas is

In the figure, a capacitor is filled with dielectrics. The resultant capacitance is,

A body is projected upwards with a velocity $u$. It passes through a certain point above the ground after time $t_1$. The time after which the body passes through the same point during the return journey is

1. 222

A spherical ball of mass m and radius r rolls without slipping on a rough concave surface of large radius R. It makes small oscillations about the lowest point. Find the time period.

The intensity of an electric field depends only on the coordinates $x$ and $y$ as follows:

$E=\frac{a(x\hat{i}+y\hat{j})}{x^2+y^2}$

where, $a$ is a constant and $\hat{i}$ and $\hat{j}$ are the unit vectors of the $x$ and $y-$ axes. Find the charge within a sphere of radius $R$ with the centre at the origin.

Two sources $S_1$ and $S_2$ emitting light of wavelengths $600\text{nm}$ are placed a distance $1\times {10}^{-2}\text{cm}$ apart. A detector can be moved on the line $S_{1}P$ which is perpendicular to $S_1{S}_2$. Find out the position of the first minimum detected.

A block of mass $5kg$ is on a rough horizontal surface and is at rest. Now a force of $24N$ is imparted to it with negligible impulse. If the coefficient of kinetic friction is $0.4$ and $g=9.8{\text{m/s}}^2$, then the acceleration of the block is

What is null point

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 sections at A and B are 4 $c{m}^2$ and 2 $c{m}^2$ respectively, find the rate of flow of water across any section.

The time for a particle executing SHM with frequency 1 Hz is considered from initial phase of $\frac{\pi }{4}$. The time at which potential energy is three times of kinetic energy is (in second):

For an ideal gas, the ratio of specific heat capacities is $\frac{C_P}{C_V}=\gamma$. If the gas undergoes a polytropic process according to the equation $P{V}^n=k$, find the range of $n$ for which the temperature of the gas increases when it rejects heat to the surroundings.

A particle starts from rest. The $a-x$ curve is shown in the figure for the motion of particle. The speed of the particle at $x=5\text{m}$ is

A boy uses a vernier callipers to measure the thickness of his pencil. He measures it to be $1.40\text{mm}$. If the zero error of vernier callipers is $+0.02\text{cm}$, what is the correct thickness of the pencil?

A block of mass $0.50\text{kg}$ is moving with a speed of $2.00{\text{ms}}^{-1}$ on a smooth surface. it strikes another mass of $1.00\text{kg}$ and then they move together as a single body. The energy loss during the collision is

The string shown in the figure is passing over a smooth pulley rigidly attached to the trolley $A$. The speed of the trolley is constant and equal to $v_A$. Speed and magnitude of acceleration of block $B$ at the instant shown in the figure is $v_B$ and $a_B$ respectively, then

An inductor of inductance $\left(L\right)$ and a resistor of resistance $\left(R\right)$ are initially connected with the battery in series to steady state and then the battery is removed. The total heat produced in the resistor $R$, when the current in the circuit decreases from maximum value $i_0$ to $0$ is -

A particle of specific charge $\alpha$ starts moving from the origin under the action of an electric field $\overrightarrow{E}=E_0\hat{i}$ and magnetic field $\overrightarrow{B}=B_0\hat{k}$. Its velocity at $(x_0,y_0,0)$ is $(4\hat{i}+3\hat{j})$. The value of $x_0$ is