Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity 'v' and other with uniform acceleration 'a'. If \[\alpha \] is the angle between the lines of motion of two particles, then the least value of relative velocity will be at time given by
A solid sphere of radius 2 m rolls without slipping on horizontal surface. Centre of mass has velocity \[{{v}_{0}}=4\text{ }m/s\]and acceleration \[10\text{ }m/{{s}^{2}}\] as shown in figure, then acceleration at point P is
Sixty-four spherical rain drops of equal size are falling vertically through air with a terminal velocity\[1.5\text{ }m{{s}^{-1}}\]. If these drops coalesce to form a big spherical drop, then the terminal velocity of big drop is
The coefficient of linear expansion of an inhomogeneous rod changes linearly from \[{{\alpha }_{1}}\] to \[{{\alpha }_{2}}\] from one end to other end of the rod. The effective coefficient of linear expansion of the rod is
The angular frequency of damped oscillator is given by \[\omega =\sqrt{\left( \frac{k}{m}-\frac{{{r}^{2}}}{4{{m}^{2}}} \right)}\] where k is the spring constant, m is the mass of the oscillator and r is the damping constant. If the ratio \[\frac{{{r}^{2}}}{mk}\] is \[8%,\] then the undamped oscillator is, approximately, as follows
Four resistors are connected as shown in following figure. A 6V battery of negligible resistance is connected across terminals A and C. The potential difference across terminals B and D will be
In a certain region of space there exists a constant and uniform magnetic field of induction B. The width of the magnetic field is a. A charged particle having charge q, is projected perpendicular to B and along the width of the field. If deflection produced by the field perpendicular to the width is d, then the magnitude of the momentum of the particle is
Every iron-atom in a ferromagnetic domain in iron has a magnetic dipole moment equal to \[9.27\times {{10}^{-24}}A/{{m}^{2}}\]. A ferromagnetic domain in iron has the shape of a cube of side \[1\mu m\]. The maximum dipole moment occurs when all the dipoles are aligned. The molar mass of iron is 56 g and its specific gravity is 8. The approximate magnetization of the domain is
Two right angled prisms having the same refracting angle \[60{}^\circ \] are placed as shown in the figure. A ray of light incident on the first prism is finally deviated by an angle \[90{}^\circ \]. The refractive index of second prism is \[\sqrt{2}\]. The refractive index of the first prism is equal to
A pencil (AB) of length 20 cm is moving along the principal axis of a concave mirror MM?. with a velocity \[5\text{ }m/s\]approaching the mirror. The mirror itself is moving away from the pencil at a speed of\[2\text{ }m/s\]. Find the rate of change of length of the image of the pencil at the instant end A is at a distance of 60 cm from the mirror.
A radioactive material of half-life T was produced in a nuclear reactor at different instants. The quantity produced second time was twice of that produced first time, if now their present activities are \[{{A}_{1}}\] and \[{{A}_{2}},\] respectively, then their age difference equals
Graph showing the electric potential V versus distance r from the centre of two concentric conducting spherical shells is as shown in the figure. The outer shell is thick and has total charge Q with inner radius 2a and outer radius 3a, the inner shell is thin having radius a and is earthed.
A particle having mass m and charge q moves along a line under the action of electric field \[E=\alpha -\beta x,\] where \[\alpha \] and \[\beta \] are positive constants and x is the distance from a point where the particle initially is at rest. Therefore for an observer moving with an acceleration \[\frac{q\alpha }{m},\]
(i) The motion of particle is oscillatory
(ii) The amplitude of particle is \[\frac{\alpha }{\beta }\]
(iii) The mean position of particle is \[x=\frac{\alpha }{\beta }\]
(iv) Maximum acceleration of particle is \[\frac{q}{\beta }\alpha \]
A ball of mass m with a charge \[+q\] can rotate in a vertical plane at the end of a string of length L in a uniform electrostatic field whose lines of forces are directed vertically upwards. The horizontal velocity that must be imparted to the ball at top position so that tension in the string at the bottom position of the ball is 15 times the weight of ball, is
A massive star is spinning about its diameter with an angular speed \[{{\omega }_{0}}=\frac{\pi }{1000}rad/day.\] After its fuel is exhausted, the star collapses under its own gravity to form a neutron star. Assume that the volume of the star decreases to \[{{10}^{-12}}\] times the original volume and its shape remains spherical. Assuming that the density of the star is uniform, find the angular speed of the neutron star.
An infinite current carrying conductor carries current i and lies parallel to Z-axis and situated at point P as shown in figure. Find \[\int\limits_{A}^{B}{\vec{B}\,.d\vec{\ell }}.\]
The position of a particle moving along a straight line is given by \[x(t)=\frac{A}{B}(1-{{e}^{At}}),\] where B is constant and\[A>0\]. The dimensions of \[\frac{{{A}^{3}}}{B}\] is same as
Two blocks of mass 2 kg and 3 kg are arranged as shown in the figure. The value of friction coefficient between 2 kg and 3 kg surface is \[0.4,\] and \[0.02t\] between the surface of 3 kg block and ground. A time varying horizontal external force \[F=5t\]is acting on 3 kg block (where t is time in sec.) Work done by the friction force on 2 kg block up to 5 sec with respect to 3 kg block is
A force F depends on displacement x as \[F=6x+4,\] where F is in Newton and x in metre, and it acts on a mass \[m=2\text{ }kg\] which is initially at rest at point \[x=0\]. Find the velocity (in m/s) of mass when \[x=2\text{ }m\] (Assume that no other force is acting on mass m.)
Two elastic balls are suspended at the same height, one has mass \[{{m}_{1}}=0.2\,kg\] and other has mass \[{{m}_{2}}\]. If the system is left alone into the position as shown in figure, we find that after an elastic head on collision both balls rise to the same height. What is the mass of the other ball in (kg)?
Two satellites A and B revolve around a planet in coplanar circular orbit in the same direction with period of revolutions 1 hour and 8 hours respectively. The radius of satellite A is \[{{10}^{4}}\] km, then the angular speed of B with respect to A in rad/hour is \[\frac{\pi }{N}\]. Find the value of N.
An inductor coil is connected to an AC source through a \[60\Omega \] resistance in series. The source voltage, voltage across the coil and voltage across the resistance are found to be \[33\text{ }V,\] \[\text{27 V}\] and 12 V respectively. What is the resistance of the coil in \[(\Omega )\]?
In a Young's double slit experiment, constructive interference is produced at a certain point P. The intensities of light at P due to the individual source are 4 and 9 units. What is the resultant intensity at point P?
Half-life of a first order reaction is 4 s and the initial concentration of the reactant is\[0.12\text{ }M\]. The concentration of the reactant left after \[16\text{ }s\] is
The azeotropic mixture of water (b.p.\[100{}^\circ C\]) and \[HCl\](b.p.\[85{}^\circ C\]) boils at \[108.5{}^\circ C\]. When this mixture is distilled it is possible to obtain
A)
Pure \[HCl\]
doneclear
B)
Pure water
doneclear
C)
Pure water as well as pure \[HCl\]
doneclear
D)
neither \[HCl\] nor \[{{H}_{2}}O\] in their pure states
If the concentration of \[O{{H}^{-}}\] ions in the reaction \[Fe{{(OH)}_{3}}(s)\rightleftharpoons F{{e}^{3+}}(aq)+3O{{H}^{-}}(aq)\] is decreased by \[\frac{1}{4}\] times, then equilibrium concentration of \[F{{e}^{3+}}\] will increase by:
In an experiment, \[4\text{ }g\] of \[{{M}_{2}}{{O}_{x}}\] oxide was reduced to \[2.8\text{ }g\] of the metal. If the atomic mass of the metal is \[56\text{ }g\text{ }mo{{l}^{-1}},\] the number of O atoms in the oxide is
An element crystallises in a structure having a fcc unit cell of an edge length 200 pm. Calculate its density if \[200\text{ }g\]of this element contains \[24\times {{10}^{23}}\]atoms?
A buffer solution is prepared by mixing \[10\text{ }mL\] of \[1.0\text{ }M\] \[C{{H}_{3}}COOH\] and \[20\text{ }mL\] of \[0.5\text{ }M\]\[C{{H}_{3}}COONa\] and then diluted to \[100\text{ }mL\] with distilled water. If \[p{{K}_{a}}\] of \[C{{H}_{3}}COOH\] is \[4.76,\] what is the pH of the buffer solution?
The distance of the point \[\hat{i}+2\hat{j}+3\hat{k}\] from the plane \[\vec{r}.(\hat{i}+\hat{j}+\hat{k})=5\] measured parallel to the vector \[2\hat{i}+3\hat{j}-6\hat{k}\] is________.