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An air bubble of radius 5mm rises through a vat of syrup at a steady speed of 4mm/s. If the syrup has a density of $1.4\times {10}^{3}kg/m^3$, what is its viscosity?

In the following figure, the pulley $P_1$ is fixed and the pulley $P_2$ is movable. If $W_1=W_2=100\text{N}$, what is the the value of $\angle A{P}_2{P}_1$? The pulleys are light and frictionless.

In order to measure physical quantities in the sub-atomic world, the quantum theory often employs energy $E$, angular momentum $J$ and velocity $c$ as fundamental dimensions instead of the usual mass, length and time. Then, the dimension of force in this theory is

Two charges $-4\mu \text{C}$ and $+4\mu \text{C}$ are placed at the points $\text{A}(1,0,4)$ and $\text{B}(2,-1,5)$ located in an uniform electric field $\overrightarrow{E}=20\hat{i}\text{N/C}$. The torque acting on the dipole has a magnitude of

A conducting rod PQ of length L = 1.0 m is moving with uniform speed v = 2.0 $m{s}^{-1}$ in a uniform magnetic field B = 4.0 T directed into the paper. A capacitor of capacity C = 10 $\mu$F is connected as shown in the figure. Then after a long time charge on plates of capacitor is

A particle of mass m is projected from the ground with an initial speed $u_0$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_0$. The angle that the composite system makes with the horizontal immediately after the collision is ?

A string of length $l=1\text{m}$ fixed at one end carries a mass $m=1\text{kg}$ at the other end. The string is rotating at a rate of $\frac{2}{\pi }\text{rev/s}$ about the axis through the fixed end as shown in the figure. Then the tension in the string is

The distance (in units) of the point $(1,-5,9)$ from the plane $x-y+z=5$ measured along the line $x=y=z$ is:

A capillary tube with inner cross-section in the form of a square of side ${}^{\prime }{a}^{\prime }$ is dipped vertically in a liquid of density $\rho$ and surface tension $\sigma$ which wets the surface of the capillary tube with an angle of contact $\theta$. The approximate height to which the liquid will rise in the tube is: (Neglect the effect of surface tension at the corners of the capillary tube)

The length of the string of a simple pendulum is measured with a metre scale to be 90.0 cm, The radius of the bob plus the length of the hook is calculated to be 2.13 cm using measurements with a slide callipers.
What is the effective length of the pendulum ? (The effective length is defined as the distance between the point of suspension and the centre of the bob.)

A block of mass m rests on an inclined plane and is attached by a string to the wall as shown in the figure. The coefficient of static friction between the plane and the block is 0.25. The string can withstand a maximum force of 20 N. The maximum value of the mass (m) for which the string will not break and the block will be in static equilibrium is kg.

Take $\cos \theta =0.8and\sin \theta =0.6$

Acceleration due to gravity $g=10m/s^2$

A message of frequency 5 kHz and peak voltage of 2 V is used to modulate a carrier wave of frequency 2 MHz and a peak voltage of 10 V. The modulation index is

Two particles move at right angles to each other. Their de-Broglie wavelengths are ${\lambda }_1$ and ${\lambda }_2$ respectively. The particles suffer perfectly inelastic collision. The de-Broglie wavelength $\lambda$, of the final particle, is given by

A force $F$ is applied to a block of mass $2\sqrt{3}\text{kg}$ as shown in the diagram. What should be the maximum value of force so that the block does not move?

The speed of sound in a mixture of $1$ mole of helium and $2$ moles of oxygen at $27{}^{\circ }\text{C}$
will be

When a plane mirror $AB$ is placed horizontally on level ground, with $B$ at a distance of $60m$ from the foot of a tower, the top of the tower and its image in the mirror subtends, an angle of ${90}^{\circ }$ at $B$. The height of the tower is

An unknown quantity $‘{\alpha }^{\prime }$ is expressed as $\alpha =\frac{2ma}{\beta }log(1+\frac{2\beta \ell }{ma})$ where m = mass, a = acceleration, $\ell =$ length. The unit of $\alpha$ should be ?

Which of the following is/are equal to $\int {\text{cosec}}^{4}xdx$

(where $C$ is integration constant)

Find the time required for a cylindrical tank of height $H$ and radius $r$ to drain half of the water through a circular hole of the area $a$ at the bottom. The flow-through hole is according to the law $v\left(t\right)=k\sqrt{2gh\left(t\right)}$, where $v\left(t\right)$ and $h\left(t\right)$ is the velocity of flow through hole and height above the hole respectively at the time $t$ and $g$ is the gravitational acceleration.

A room is maintained at ${20}^{o}C$ by a heater of resistance 20 ohm connected to 200 volt mains. The temperature is uniform through out the room and heat is transmitted through a glass window of area $1m^2$ and thickness 0.2 cm. What will be the temperature outside? Given that thermal conductivity K for glass is 0.2 cal /m/ ${}^{0}C$ /sec and J = 4.2 J/cal

A magnetic needle lying parallel to a magnetic field requires W units of work to turn it through ${60}^{\circ }$. The torque needed to maintain the needle in this position will be

Two polaroids $P_1$ and $P_2$ are placed with their axis perpendicular to each other. Unpolarised light $I_0$ is incident on $P_1$. A third polaroid $P_3$ is kept in between $P_1$ and $P_2$ such that its axis makes an angle ${45}^o$ with that of $P_1$. The intensity of transmitted light through $P_2$ is

In a concave mirror experiment, an object is placed at a distance x₁ from the focus and the image is formed at a distance x₂ from the focus. The focal length of the mirror would be

The moment of inertia of a thin rod of mass $M$ and length $L$ about an axis passing through the point at a distance $\frac{L}{4}$ from one of its end and perpendicular to the rod is

A proton, a deuteron and an $\alpha -$particle having the same kinetic energy enters a region of uniform magnetic field, moving at right angles to the magnetic field $B$. What is the ratio of the radii of their circular paths?

Two friends A and B are standing a distance x apart in an open field and wind is blowing from A to B. A beats a durm and hears the sound $t_1$ time after he sees the event. A and B interchange their positions and the experiment is reoeated . This time B hears the drum $t_2$ time after be sees the event. Calculate the velocity of sound in still Air v and the velocity of wind u. Neglect the time light takes in travelling between the friends.

An ideal monoatomic gas undergoes the cyclic process $\text{ABCDA}$ as shown in the figure. The efficiency of the cycle is

A linear harmonic oscillator of force constant $2\times {10}^6\text{N/m}$ and amplitude $0.01\text{m}$ has a total mechanical energy of $160$ Joules. Which of the following is true?

What is the moment of inertia of the rod, of mass $M$, about the axis shown in the figure?

Four rods of same material having the same cross section and length have been joined as shown. The temperature of the junction of the rods will be