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A particle is constrained to move in a circle of radius $300\text{cm}$. Its linear speed increases with time as $v=2t$, where $t$ is in second and $v$ is in $\text{m/s}$. The acceleration of the particle at $t=3\text{s}$ is

A homogeneous rod AB of length L and mass M is pivoted at the centre O in such a way that it can rotate freely in the vertical plane. The rod is initially in the horizontal position. An insect S of the same mass M falls vertically with speed V on the point C mid-way between O and B. The initial angular velocity $\omega$ in terms of V and L is

Two equal resistance are connected in the gaps of a metre bridge if the resistance in the left is increased by 10% the balancing point shifts

1.10% to right

2.10% to left

3.9.6%to right

4.4.8% to right

The length of an elastic rod is $a$ when tension is 4 N and $b$ when the tension is 5N. The length of the rod when tension is 9N is

A square wire frame of size $L$ is dipped in a liquid. On taking out, a membrane is formed. If the surface tension of liquid is $T$, then force acting on a frame will be

A student is standing at a distance of 50 m from the bus as soon as the bus begins its motion with an acceleration of 1 M per second the stuudent starts running towards the Bus with uniform velocity U asuming the motion to be along straight road the minimum value of u so that the student is able to catch the bus is

The value of $\frac{\left(\frac{10!}{5!}\right)}{10}$ is

A wooden block is placed on an inclined plane. Angle of inclination is continuously changed. At ${45}^{\circ }$ Blockstarts to slick and continues uniform speed if angle of inclination is reduced to ${30}^{\circ }$. Find ${\mu }_s$ and ${\mu }_k$

Find the value of $|\overrightarrow{a}\times \overrightarrow{b}{|}^2.$

A uniform magnetic field is directed out of the page. A charged particle, moving in the plane of the page, follows a clockwise spiral of decreasing radius as shown, then

A rod of length l and mass m is hanged from one edge. The time period of small oscillations is

A ray of light travelling in a medium of refractive index $\mu$ is incident at an angle $\theta$ on a composite transparent plate consisting of 50 plates of $R.I.1.01\mu ,1.02\mu ,1.03\mu$ , ........ , $1.50\mu$ . The ray emerges from the composite plate into a medium of refractive index $1.6\mu$ at angle x . Then:

Consider a spherical shell of radius $R$ at temperature $T$. The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume $u=\frac{U}{V}\propto T^4$ and pressure $P=\frac{1}{3}\left(\frac{U}{V}\right)$ . If the shell now undergoes an adiabatic expansion the relation between $T$ and $R$ is

Electrons of mass $m$ with de-Broglie wavelength $\lambda$ fall on the target in an X-rays tube. The cutoff wavelength $\left({\lambda }_0\right)$ of the emitted X-rays is

Two blocks A and B of mass m and 2m connected by a light spring of spring constant lie at rest on a fixed smooth horizontal plane are given velocities initially of magnitudes 2u and u as shown in figure. In the subsequent motion, the only horizontal force acting on blocks is due to spring. Match the conditions in column I with the instants of time they occur as given in column II.

$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{(A)The length of spring is least at time}& \text{(p)}\frac{\pi }{2}\sqrt{\frac{2m}{3k}}\\ \text{(B) The length of spring is maximum at time}& \text{(q)}\pi \sqrt{\frac{2m}{3k}}\\ \text{(C) The acceleration of both blocks is}& \text{(r)}\pi \sqrt{\frac{3m}{2k}}\\ \text{zero simultaneously at time}& \\ \text{(D) velocity of center of mass at time t = 0 is}& \text{(s) zero}\end{array}$

Choose the correct option

A 160 watt infrared source is radiating light of wavelength 50000Å uniformly in all directions. The photon flux at a distance of 1.8m is of the order of

Obtain an expression for the capacitance of a parallel plate capacitor with air between the plates.

Three identical metal balls each of radius "R” are placed on a horizontal plane touching each other such that their centres form the vertices of an equilateral triangle. The distance of the centre of mass of the system from the centre of any one of the balls is

Match the items in the second list with corresponding items from the first list

$\begin{array}{cccc}& ListI& & ListII\\ \left(A\right)& Variationofgwithheight\left(foranyheight\right)& e& g^{\prime }=g(1-\frac{2h}{R})\\ \left(B\right)& Variationofgwithheight(forsmallheightsh<<R)& f& \frac{GM}{R^2}\\ \left(C\right)& Variationofgwithdepth& g& g^{\prime }=\frac{g{R}^2}{(R+h{)}^2}\\ \left(D\right)& gonthesurfaceoftheplanet& h& g^{\prime }=g(1-\frac{d}{R})\end{array}$

The minimum angular momentum of electron of hydrogen atom in stationary orbit is

A uniform disc of radius $R=0.2\text{m}$ kept over a rough horizontal surface is given a translational velocity $V_0$ and angular velocity ${\omega }_0$. After some time, its kinetic energy become zero, then value of ${\omega }_0$ is:
Given: $V_0=10\text{m/s}$.

The potential energy (in joule) of a body of mass 2 kg moving in the x-y plane is given by
U = 6x + 8y
where the position coordinates x and y are measured in metre. If the body is at rest at point (6 m, 4m) at time t = 0, it will cross the y-axis at time t equal to

In the absorption spectrum, the wavelengths which are absorbed are missing and they appear as

An unpolarised beam of light is made to incident on an arrangement of four polarsing sheets arranged in such a way that the characteristic direction of each polarizing sheet makes an angle of ${45}^{\circ }$ with that of preceding sheet. What fraction of incident unpolarised light will be transmitted finally?

A pebble of mass 0.05 kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble,
a) during its upward motion,
b) during its downward motion,
c) at the highest point where it is momentarily at rest. Do your answers change if the pebble was thrown at an angle of ${45}^{\circ }$ with the horizontal direction?
Ignore air resistance.

If the frequency of the $2^{nd}$ and $5^{th}$ harmonics of string fixed at both ends differ by $108\text{Hz}$. What is the fundamental frequency of the string?

Which of the following is/are necessary and sufficient condition(s) for pure translation?

What is equation of a line passing through the points $(4,2)$ and $(15,-4)$?

Two containers of equal volume contain the same gas at pressures $P_1$ and $P_2$ and absolute temperatures $T_1$ and $T_2$ respectively. On joining the vessels, the gas reaches a common pressure P and common temperature T. The ratio $\frac{P}{T}$ is equal to

A steel scale measures the length of a copper rod as L cm when both are at ${20}^{\circ }C$, the calibration temperature for the scale. If the coefficients of linear expansion for steel and copper are ${\alpha }_s$ and ${\alpha }_c$ respectively, what would be the scale reading (in cm) when both are at ${21}^{\circ }C$?

A gas mixture consists of 2 moles of oxygen and 4 moles argon at temperature T. Neglecting all vibrational modes, the total internal energy of the system is

Maximum density of H2O is at the temperature

F = $K\times m\times \frac{(V-U)}{t}$; where K is a constant, the equation can be rewritten as:

A body moves a distance of 10 m along a straight line under the action of a force of 5 N. If the work done is 25 joule, the angle which the force makes with the direction of motion of the body is

A mass is supported on a frictionless horizontal surface. It is attached to a string and rotates about a fixed centre at an angular velocity ${\omega }_0$. If the length of the string and angular velocity are doubled, the tension in the string which was initially $T_0$ is now

A motor car moving with a uniform speed of $20m{s}^{-1}$ comes to rest on the application of brakes after travelling a distance of 10 m. Its acceleration is

Consider a satellite going round the earth in an orbit. Which of the following statements is wrong

Find the charges on the four capacitors of capacitances 1$\mu F,2\mu F,3\mu F$ and $\mu F$ shown in figure (32 - E32). Figure 32 - E32

Figure 32 - E32

A cubical block of ice of mass 'm' and edge 'L' is placed in a large tray of mass 'M'. If the ice melts, how far does the centre of mass of the system "ice plus tray” come down with reference to the point O as shown in figure

Three identical dipoles are arranged as shown below. What will be the net electric field at P? $(K=\frac{1}{4\pi {ϵ}_0})$