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A stone is tied to a rope of length $2$ m and is whirled above his head horizontally by a man at a height of $9\text{m}$ from the ground. The stone covers a horizontal distance of $20\text{m}$ when the rope breaks. What is the magnitude of the centripetal acceleration when the stone was in the circular motion?

A particle moves along fixed line (x axis) as follows:
$X=6\cos (\frac{4\pi }{3}t+\frac{\pi }{6})$ mm.
Find time when particle attains maximum speed for first time.

A circular ring is expanding in a uniform magnetic field of 2 T, which is perpendicular to the plane of the ring such that, its radius is increasing at a rate of 2 cm/s. Induced EMF in the ring when radius is 40 cm is

A charged particle with some initial velocity is projected in a region where non-zero electric and/or magnetic fields are present. In column I, information about the existence of electric and/or magnetic field and direction of initial velocity of charged particle are given; while in Column II, the probable path of the charged particle is mentioned. Match the entries of Column I with the entries of Column II.
$\begin{array}{cccc}& \text{Colum I}& & \text{Colum II}\\ \text{i.}& \overrightarrow{E}=0,\overrightarrow{B}\ne 0\text{and initial velocity may}& \text{a.}& \text{Straight line}\\ & \text{be at any angle with}\overrightarrow{B}.& & \\ \text{ii.}& \overrightarrow{E}=0,\overrightarrow{B}=0\text{and initial velocity may}& \text{b.}& \text{Parabola}\\ & \text{be at any angle with}\overrightarrow{E}.& & \\ \text{iii}& \overrightarrow{E}\ne 0,\overrightarrow{B}\ne 0,\overrightarrow{E}||\overrightarrow{B}\text{and initial velocity is}& \text{c.}& \text{Circular}\\ & \perp \text{to both.}& & \\ & & \text{d.}& \text{Helical path}\end{array}$

A coil having an area $A_0$ is placed in a magnetic field which changes from $B_{0}to3B_0$ in a time interval t. The e.m.f. induced in the coil will be

Two soap bubbles of radii $r_{1}and{r}_2$ equal to 4cm and 5cm are touching each other over a common surface $S_1{S}_2$ (shown in figure). The radius of common surface will be

A 10 g bullet having a charge of 4.00 $\mu C$ is fired at a speed of 270 m $s^{-1}$ in a horizontal direction. A vertical magnetic field of 500 $\mu T$ exists in the space. Find the deflection of the bullet due to the magnetic field as it travels through 100 m. Make appropriate approximations.

A calorimeter of negligible heat capacity contains 100 cc of water at ${40}^0$ C. The water cools to ${35}^0$ C in 5 minutes. The water is now replaced by K-oil of equal volume at ${40}^0$ C. Find the time taken for the temperature to become ${35}^0$ C under similar conditions. Specific heat capacities of water and K-oil are 4200 J $k{g}^{-1}{K}^{-1}$ and 2100 J $k{g}^{-1}{K}^{-1}$ respectively. Density of K-oil = 800 kg $m^{-3}$.

If the current $3.0A$ flowing in the primary coil is made zero in $0.1sec$, the emf induced in the secondary coil is $1.5V$. The mutual inductance between the coils is

Three capacitors each of capacitance 1μF are connected in parallel. To this

combination, a fourth capacitor of capacitance 1μF is connected in series.

The resultant capacitance of the system is

The cylindrical tube of a spray pump has radius R, one end of which has n fine holes, each of radius r. If the speed of the liquid in the tube is V, the speed of the ejection of the liquid through the holes is:

In a YDSE apparatus, the source (wavelength $\lambda$) is kept as shown in figure. None of the slits receives direct light but get it after reflection by mirrors as shown in the figure. If $d=100\lambda$ and $D=50000\lambda$. Then the distance of central maxima from the center of the screen

Find the electric field vector at P(a,a,a) due to three infinitely long lines of charges along the x, y and z axis, respectively. The charge density i.e. charge per unit length of each wire is $\mathrm{\lambda .}$

A bomb is dropped from a plane flying horizontally with uniform speed. Show that the bomb will explode vertically below the plane. Is the statement true if the plane files with uniform speed but not horizonrally ?

Two point A and B Lie on two extremes of a diameter AB as shown in the figure. The sphere rotates with constant angular velocity $\omega$ about an axis as shown. What is the angular velocity of Q with respect to P?

A conducting wire of length l, lying normal to a magnetic field B, moves with a velocity v as shown in figure (34 - E11). (a) Find the average magnetic force on a free electron of the wire. (b) Due to this magnetic force, electrons concentrate at one end resulting in an electric field inside the wire. The redistribution stops when the electric force on the free electrons balances the magnetic force. Find the electric field developed inside the wire when the redistribution stops. (c) What potential difference is developed between the ends of the wire ?

Two beads of mass 2m and m, connected by a rod of length l of negligible mass are free to move in a smooth vertical circular wire frame of radius l as shown. Initially the system is held in horizontal position (as shown in the figure). The velocity that should be given to mass 2m (when rod is in horizontal position) in counter-clockwise direction so that the rod can just become vertical is :

A bubble of air has radius $0.2\text{mm}$ in water. If the bubble had been formed $20\text{cm}$ below the water surface on a day when the atmospheric pressure was $1.013\times {10}^5\text{Pa}$, then what would have been the total pressure inside the bubble? (surface tension of water = $73\times {10}^{-3}\text{N/m}$)

A particle is subjected to two simple harmonic motions of same time period in the same direction. The amplitude of the first motion is 3.0 cm and that of the second 4.0 cm. Find the resultant amplitude if the phase difference between the motions is (a) $0^{\circ },$ (b) ${60}^{\circ },$ (c) ${90}^{\circ }.$

Given the magnitude of vector A to be 6 units. Find the component of A along the x-axis?

Two masses, 'n*m' and 'm', start simultaneously from the intersection of two straight lines with velocities 'v' and 'n*v' respectively. It is observed that the path of their centre of mass is a straight line bisecting the angle between the given straight lines. Find the magnitude of the velocity of centre of mass.[Here $\theta$ = angle between the lines]

When a sound wave of wavelength $\lambda$ is propagating in a medium, the maximum velocity of the particle is equal to the wave velocity. The amplitude of wave is:

A body of mass 2 kg is lying on a rough inclined plane of inclination ${30}^{\circ }$.. Find the magnitude of the force parallel to the incline needed to make the block move (a) up the incline (b) down the incline, coefficient of static friction = 0.2

A frictionless track $ABCDE$ ends in a circular loop of radius $R$. A body slides down the track from point $A$ which is at a height $h=5\text{cm}$. Maximum value of $R$ for the body to successfully complete the loop is

In the relation $P=\frac{\alpha }{\beta }{e}^{\frac{\alpha x}{nR\theta }}$, P is power, x is distance, n is number of moles, R is gas constant and $\theta$ is temperature. The dimensional formula of $\beta$ is

A thin spherical shell of radius R lying on a rough horizontal surface is hit sharply and horizontally by a cue. At what height from the ground should it be hit so that the shell does not slip on the surface:

A projectile is thrown with a velocity of $50m{s}^{-1}$ at an angle of ${53}^o$ with the horizontal. The equation of the trajectory is given by (Take $g=10m/s^2$)

Light is incident normally on face AB of a prism as shown in figure. A liquid of refractive index μ is placed on face AC of the prism. The prism is made of glass of refractive index 3/2 . The limits of 'μ'for which total internal reflection takes place on face AC is

A highly rigid cubical block $A$ of small mass $M$ and side $L$ is fixed rigidly on to another cubical block $B$ of the same dimensions and of low modulus of rigidity η such that the lower face of $A$ completely covers the upper face of $B$. The lower face of $B$ is rigidly held on a horizontal surface. A small force $F$ is applied perpendicular to one of a sides faces of $A$. After the force is withdrawn, block $A$ executes small oscillations. The time period of oscillation is

Two blocks of masses $m_1$ and $m_2$ are connected by a light inextensible string which passes over a fixed pulley. Initially, mass $m_1$ moves with a velocity $v_0$ when the string is not taut. Neglect friction at all contact surfaces. Find the velocity of the blocks just after the string is taut, if $m_1=2\text{kg},m_2=3\text{kg}$ and $v_0=5\text{m/s}$

A particle falls from a height h upon a fixed horizontal plane and rebounds. If e is the coefficient of restitution, the total distance travelled before rebounding has stopped is

Figure shows a particle sliding on a frictionless track which terminates in a straight horizontal section. If the particle starts slipping from the point A, how far away from the track will the particle hit the ground?

For a particle moving along $x$ - axis, the acceleration $a$ of the particle in terms of its $x$ - coordinate is given by $a=-9x$, where $x$ is in meters and $a$ is in ${\text{m/s}}^2$. Take acceleration, velocity and displacement in positive $x$ - direction as positive. The initial velocity of particle at $x=0$ is $u=+6\text{m/s}$. The maximum distance of the particle along the positive $x$-direction from origin will be

A ball of mass $500\text{gm}$ is tied to one end of a rope of length $3\text{m}$. The other end is fixed and the ball is now moving along a horizontal circle with uniform angular speed of $2\text{rev/s}$. The tension in the rope (in $\text{newton}$) will be:

A horse is harnessed to a cart. If the horse tries to pull the cart, the horse must exert a force on the cart. By Newton's third law the cart must then exert an equal and opposite force on the horse. Since the two forces are equal and opposite, they must add to zero, so Newton's second law tells us that the acceleration of the system must be zero and therefore no matter how hard the horse pulls, it can never move the cart.

The radius of the first Bohr orbit of a hydrogen atom is $0.53\times {10}^{-8}cm.$ The velocity of the electron in the first Bohr orbit is:

You have but one molecule of Hydrogen $\left(H_2\right),$ meandearing about in the vast emptiness of a $1c{m}^3$ closed box, at a speed of 500 m/s, bouncing off from one wall to the next. What are the rms speed and the temperature inside the box?

Find the decrease in the volume of a sample of water from the following data. Initial volume = 1000 cm³, initial pressure = 10⁵ N m^-2, final pressure = 10⁶ N m^-2, compressibility of water = 50 x 10^-11 m² N^-1.

How to convert 4800 angstroms to m

Two bodies, a sphere and a cube made of the same material and having the same surface areas are heated to the same temperature and allowed to cool in the same surroundings. The ratio of the amount of radiation emitted in equal time intervals will be

A force $F=2\text{N}$ acts at a distance $x=0.5\text{m}$ above the centre of a thin spherical shell of mass $m=3\text{kg}$ and radius $R=2\text{m}$. Find out the linear acceleration $\left(a\right)$ and angular acceleration $\left(\alpha \right)$ of the shell.

the energy of first exited state of he+ atom is