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A string of length $L$ is fixed at one end and carries a mass $M$ at the other end. The string makes $\frac{2}{\pi }$ revolutions per second around the vertical axis through the fixed end as shown in the figure, then tension in the string is

If $y=\tan \left[\text{log}\right(x^2\left)\right]$, then $\frac{dy}{dx}$ is -

The magnitude of a given vector with end points (4, -4, 0) and (-2, -2, 0) must be

Two waves are propagating to the point P along a straight line produced by two sources A and B of simple harmonic and of equal frequency. The amplitude of every wave at P is 'a' and the phase of A is ahead by than that of B and the distance AP is greater than BP by 50 cm. Then the resultant amplitude at the point P will be, if the wavelength is 1 meter

[BVP 2003]

In the arrangement shown in figure find the tensions in the rope and accelerations of the masses $m_1$ and $m_2$ and pulleys $P_1$ and $P_2$ when the system is set free to move. Assume the pulleys to be massless and strings are light and inextensible.

p) T (i) m₁g

q) a₁ (ii) m₂g

r) a₂ (iii) (m₁ - m₂)g

s) ap₁ (iv) g downward

t) ap₂ (v) g upward

(vi) 3g downward

A 5 cm thick ice block is there on the surface of water in a lake. The temperature of air is -10 degree Celsius, how much time it will take to double the thickness of the block..,..

A small metal plate (work function $\phi$)is kept at a distance d from a singly ionized, fixed ion. A monochromatic light beam is incident on the metal plate and photo electrons are emitted. Find the maximum wavelength of the light beam so that some of the photo electrons may go round the ion along a circle.

According to Bohr's theory the radius of electron in an orbit described by principal quantum number n and atomic number Z is proportional to

A thin but long, hollow, cylindrical tube of radius r carries a current i along its length. Find the magnitude of the magnetic field at a distance $r/2$ from the surface (a) inside the tube (b) outside the tube.

A uniform square lamina of side 2a is hung up by one corner and oscillates in its own plane which is vertical. Find the length of the equivalent simple pendulum.

A linear charge having linear charge density $\lambda$ penetrates a cube diagonally and then it penetrates a sphere diametrically as shown. What will be the ratio of flux coming out of cube and sphere?

A satellite of mass $1000kg$ is rotating around the earth in a circular orbit of radius $3R$. What extra energy should be given to this satellite, if it is to be lifted into an orbit of radius $4R$?

A wooden block of mass $20g$ is dropped from the top of the cliff $50m$ high. Simultaneously a bullet of mass $20g$ is fired from the foot of the cliff upwards with a velocity $25m{s}^{-1}$. The bullet and the wooden block will meet each other after time : ($g=10m{s}^{-2}$)

A torch bulb rated as $4.5W,1.5V$ is connected as shown in the figure. The e.m.f. of the cell(in Volts) needed to make the bulb glow at full intensity is

A block of mass m = 15 kg is suspended in an elevator with the help of three identical light elastic cords (spring constant k = 100 N/m each) attached vertically. Cord 1 is tied to the ceiling of the elevator and the other two cords 2 and 3 are tied to the elevator floor as shown in the figure. When the elevator is stationary the tension force in each of the lower cords is T = 7.5 N. Take g = 10 $m/s^2$. If now the elevator starts moving with given four accelerations shown in column I and Column II gives the displacement of the block with respect to elevator when it is accelerating. Column III gives tension in the cord 1 when elevator is accelerating.

Column III represents the tension in chord 1.

One mole of an ideal gas undergoes a process in which $T=T_0+a{V}^3$, where $T_0$ and ‘a’ are positive constants and V is the volume. The volume for which pressure will be minimum is

The x - z plane is the boundary between two transparent media. Medium 1 with $z\ge 0$ has a refractive index $\sqrt{2}$ and medium 2 with $z\le 0$ has a refractive index $\sqrt{3}$. A ray of light in medium 1 given by the vector $A=6\sqrt{3}\hat{i}+8\sqrt{3}\hat{j}-10\hat{k}$ is incident on the plane of separation. The angle between vector A and the positive z - direction is

A block of mass $m$ is moving with an initial velocity $v_0$ towards a stationary spring of stiffness $k$ attached to the wall as shown. Find the maximum compression in the spring.

An insect crawls up a hemispherical surface very slowly (see figure below). The coefficient of friction between the insect and the surface is $\frac{1}{3}$. If the line joining the centre of hemispherical surface to the insect makes an angle $\alpha$ with the vertical, the maximum possible value of $\alpha$ is given by

Three particles of masses $1\text{kg}$ , $2\text{kg}$ and $3\text{kg}$ are placed at the vertices $A,B,C$ respectively of an equilateral triangle $ABC$ of edge $1\text{m}$ as shown in the figure. Find the position of the centre of mass of the system. (If $A$ is assumed to be the origin).

Two particles $A$ and $B$ moving in $x-y$ plane are at origin at $t=0\text{sec}$.
The initial velocity vectors of $A$ and $B$ are ${\overrightarrow{u}}_A=8\hat{i}\text{m/s}$ and ${\overrightarrow{u}}_B=8\hat{j}\text{m/s}$.
The acceleration of $A$ and $B$ are constant and are ${\overrightarrow{a}}_A=-2\hat{i}{\text{m/s}}^2$ and ${\overrightarrow{a}}_B=-2\hat{j}{\text{m/s}}^2$.
List-I gives certain statements regarding particle $A$ and $B$.
List - II gives corresponding results in SI units.
Match the statements in List-I with corresponding results in List-II.
$\begin{array}{cccc}& \text{List-I}& & \text{List-II}\\ \text{(I)}& \text{The time (in seconds) at which velocity of A relative}& \text{(P)}& 16\sqrt{2}\\ & \text{to B is zero}& & \\ \text{(II)}& \text{The distance (in metres) between A and B when their}& \text{(Q)}& 8\sqrt{2}\\ & \text{relative velocity is zero}& & \\ \text{(III)}& \text{The magnitude of relative velocity of A and B at the}& \text{(R)}& 8\\ & \text{instant they are at same position}& & \\ & & \text{(S)}& 4\end{array}$

A small bob of mass $m$ is suspended by a massless string from a cart of the same mass $m$ as shown in the figure. The friction between the cart and horizontal ground is negligible. The bob is given a velocity $v_0$ in horizontal direction as shown. The maximum height attained by the bob is
(initially whole system (bob+string+cart) was at rest).

An EM wave from air enters a medium. The electric fields are ${\overrightarrow{E}}_1=E_{01}\hat{x}\cos \left[2\pi v(\frac{z}{c}-t)\right]$ in air and

${\overrightarrow{E}}_2=E_{02}\hat{x}\cos \left[k\right(2z-ct\left)\right]$ in medium, where the wave number $k$ and frequency v refer to their values in air. The medium is non-magnetic. If ${\epsilon }_{r_1}$ and ${\epsilon }_{r_2}$ refer to relative permitivities of air and medium respectively, which of the following options is correct?

A concave mirror of focal length 100 cm is used to obtain the image of the

sun which subtends an angle of 30⁰ The diameter of the image of the sun

will be

A particle of charge -q & mass m moves in a circle of radius r around an infinitely long line charge having linear charge density +λ. Then time period will be--

A steel wire is rigidly fixed at both ends. Its length, mass and cross-sectional area are $1\text{m, 0.1 kg and}{10}^{-6}{\text{m}}^2$respectively. Tension in the wire is produced by lowering the temperature by ${20}^{\circ }\text{C}$. If transverse waves are set up by plucking the wire at $0.25\text{m}$ from one end, and assuming that the wire vibrates with minimum number of loops possible for such a case. The frequency of vibration (in $Hz$) is found to be $11\times \text{K}$. Find the value of $K$.$(\text{Given}\alpha =1.21\times {10}^{-5\circ }{C}^{-1},Y=2\times {10}^{11}N/m^2)$

The total acceleration of a particle in circular motion is $5m/s^2$. If the speed of the particle decreases at a rate of $3m/s^2$, assuming $r=2m$, the angular speed $\omega$ of the particle relative to the centre of the circle is

A body is projected at time t = 0 with a velocity u at an angle $\theta$ with the horizontal. The horizontal and vertical components of its velocity will become equal at a time t $(t\ne 0)$ if,

A rod of length 1m and mass 5 kg is attached to two springs as shown in figure in vertical plane. The rod is hinged about H point. If there is small angular displacement of rod from vertical position forcing it to do oscillation about H with angular frequency $\omega$, then value of [$\omega$] is : [Given K = 100 N/m and [.] is greatest integer function] ___

A capacitor, an inductor and an electric bulb are connected in series to an AC supply of variable frequency. As the frequency of the supply is increased gradually from zero, then the brightness of the electric bulb is found to

A ring of radius r is moving at constant velocity v along axis of a circular loop. If current in loop is i and radius of loop is R where R>>r, then emf induced in ring when x>>R is

A child has near point at 10 cm. What is the maximum angular magnification the child can have with a convex lens of focal length 10 cm?

A ball is dropped from a height of $30\text{m}$ in downward direction on the stationary floor. If the value of coefficient of restitution is $0.6$ and ball rebounds back up to the height $\left(h^{\prime }\right)$, find the height $\left(h^{\prime }\right)$ after first collision.
(Take $g=10{\text{m/s}}^2$)

A force of $5kg-wt$ is acting on a body of mass $5kg$ kept on a smooth horizontal surface at rest. If it acts for $10$ seconds at an angle of ${60}^{\circ }$ with the horizontal, find the distance travelled by the body. (Take $g=10m/s^2)$

An open organ pipe has a length of 5 cm. (a) Find the fundamental frequency of vibration of this pipe. (b) What is the highest harmonic of such a tube that is in the audible range ? Speed of sound in air is $340m{s}^{-1}$ and the audible range is 20-20,000 Hz.

Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time $t_1$. On other days, if she remains stationary on the moving escalator, then the escalator takes her up in time $t_2$. The time taken by her to walk up on the moving escalator will be

Two projectiles are projected with same speed at angles $(\frac{\pi }{4}+\theta )$ and $(\frac{\pi }{4}-\theta )$ with the horizontal, where $\theta <\frac{\pi }{4}$ . The ratio of horizontal ranges described by them is

A planet in a distant solar system is 10 times more massive than the earth and its radius is 10 times smaller. Given that the escape velocity from the earth is $11{\text{kms}}^{–1}$, the escape velocity from the surface of the planet would be

A car has to move on a level turn of radius $45\text{m}$. If the coefficient of static friction between the tyres and the road is ${\mu }_s=0.5$, find the maximum speed the car can achieve without skidding.

An object is thrown horizontally with a velocity of $10\text{m/s}$. Find the radius of curvature of its trajectory $3\text{s}$ after the motion has begun. Take $g=10{\text{m/s}}^2$

Two cars P and Q move with velocities ${\overrightarrow{v}}_P$ and ${\overrightarrow{v}}_Q{\overrightarrow{v}}_Q$ towards north and east respectively.

Velocity of car P w.r.t. Q (i.e., velocity of car P for an observer on cars Q) is ${\overrightarrow{v}}_{AB}$ will be in which direction?

Is anything faster than light ?

A battery of emf ${\epsilon }_0=10V$ is connected across a $1\text{m}$ long uniform wire having resistance $10\frac{\Omega }{m}$. Two cells of emf ${\epsilon }_1=2V$ and ${\epsilon }_2=4V$ having internal resistances $1\Omega$ and $5\Omega$ respectively are connected as shown in the figure. If a galvanometer shows no deflection at the point $P$, find the distance of point $P$ from the point $A$.

What is the difference between celcius and centigrade