(a) What is Brownian motion ? Draw a diagram to show the movement of a particle (like a pollen grain) during Brownian motion.
(b) In a beam of sunlight entering a room, we can sometimes see dust particles moving in a haphazard way in the air. Why do these dust particles move ?

The French physicist Louis de Broglie in 1924 postulated that matter , like radiation , should exhibit dual behaviour. He proposed the following relationship between the wavelenght lambda of a material particle , its linear momentum p and planck cosntant h. lambda= (h)/(p)=(h)/(mv) The de Broglie relation implies that the wavelength of a partices should decreases as its velocity increases. It also implies that the for a given velocity heavir particule should have shorter wavelenght than lighter particles. The waves associated with particles in motin are called matter waves or de Broglie waves. These waves differ from the electromagnetic waves as they (i) have lower velocities have no electrical and magnetic fields and are not emitted by the particle under consideration. The experimental confirmation of the de Broglie relation was obtained when Davisson and Germer. in 1927, observed that a beam of electrons is diffracted by a nickel arystal. As diffarceted by a nickel . As diffraction is a characteristic property of waves, hence the beam of electron dehaves as a wave, as proposed by de Broglie. Werner Heisenberg cobnsiderd the imits of how precisely we can measure propoerties of an electron or other microscopic particle like electron. he determined that there is accurately we measure the momentum of a particle, the less accurately we can determine its position . The converse is laso true . The is summed up in what we now call the "Hesienberg uncertainty princple: It isimpossibble to determine simultameously ltbr. and percisely both the momentum ans position of particle . The product of uncertainly in the position, Deltax and the ncertainly in the momentum Delta(mv) mudt be greater than or equal to (h)/(4pi) i.e. etaDeltax Delta(mv)ge(h)/(4pi) The correct order of wavelenght of Hydrogen (._(1)H^(1)) Deuterium (._(1)H^(2)) and Tritium (._(1)H^(3)) moving withsame kinetic energy is.

When a particle of mass m moves on the x-axis in a potential of the form V(x) =kx^(2) it performs simple harmonic motion. The correspondubing time period is proprtional to sqrtm/h , as can be seen easily using dimensional analusis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x=0 in a way different from kx^(2) and its total energy is such that the particle does not escape toin finity. Consider a particle of mass m moving on the x-axis. Its potential energy is V(x)=ax^(4)(agt0) for |x| neat the origin and becomes a constant equal to V_(0) for |x|impliesX_(0) (see figure). If total energy of the particle is E, it will perform perildic motion only if.

The velocity time graph for two particles (1 and 2) moving along X axis is shown in fig. At time t = 0 , both were at origin. (a) During first 4 second of motion what is maximum separation between the particles? At what time the separation is maximum? (b) Draw position (x) vs time (t) graph for the particles for the given interval.

When a particle is mass m moves on the x- axis in a potential of the from V(x) = kx^(2) , it performs simple harmonic motion. The corresponding thime periond is proportional to sqrt((m)/(k)) , as can be seen easily asing dimensional analysis. However, the motion of a pariticle can be periodic even when its potential enem increases on both sides x = 0 in a way different from kx^(2) and its total energy is such that the particel does not escape to infinity. consider a particle of mass m moving onthe x- axis . Its potential energy is V(x) = omega (alpha gt 0 ) for |x| near the origin and becomes a constant equal to V_(0) for |x| ge X_(0) (see figure) If the total energy of the particle is E , it will perform is periodic motion why if :

When a particle is mass m moves on the x- axis in a potential of the from V(x) = kx^(2) , it performs simple harmonic motion. The corresponding thime periond is proportional to sqrt((m)/(k)) , as can be seen easily asing dimensional analysis. However, the motion of a pariticle can be periodic even when its potential enem increases on both sides x = 0 in a way different from kx^(2) and its total energy is such that the particel does not escape to infinity. consider a particle of mass m moving onthe x- axis . Its potential energy is V(x) = omega (alpha gt 0 ) for |x| near the origin and becomes a constant equal to V_(0) for |x| ge X_(0) (see figure) If the total energy of the particle is E , it will perform is periodic motion why if :

When a particle is mass m moves on the x- axis in a potential of the from V(x) = kx^(2) , it performs simple harmonic motion. The corresponding thime periond is proportional to sqrt((m)/(k)) , as can be seen easily asing dimensional analysis. However, the motion of a pariticle can be periodic even when its potential enem increases on both sides x = 0 in a way different from kx^(2) and its total energy is such that the particel does not escape to infinity. consider a particle of mass m moving onthe x- axis . Its potential energy is V(x) = omega (alpha gt 0 ) for |x| near the origin and becomes a constant equal to V_(0) for |x| ge X_(0) (see figure) If the total energy of the particle is E , it will perform is periodic motion why if :

When a particle is mass m moves on the x- axis in a potential of the from V(x) = kx^(2) , it performs simple harmonic motion. The corresponding thime periond is proportional to sqrt((m)/(k)) , as can be seen easily asing dimensional analysis. However, the motion of a pariticle can be periodic even when its potential enem increases on both sides x = 0 in a way different from kx^(2) and its total energy is such that the particel does not escape to infinity. consider a particle of mass m moving onthe x- axis . Its potential energy is V(x) = alpha *x^4 (alpha gt 0 ) for |x| near the origin and becomes a constant equal to V_(0) for |x| ge X_(0) (see figure) If the total energy of the particle is E , it will perform is periodic motion why if :

When a particle of mass m moves on the x-axis in a potential of the form V(x) =kx^(2) it performs simple harmonic motion. The correspondubing time period is proprtional to sqrtm/h , as can be seen easily using dimensional analusis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x=0 in a way different from kx^(2) and its total energy is such that the particle does not escape toin finity. Consider a particle of mass m moving on the x-axis. Its potential energy is V(x)=ax^(4)(agt0) for |x| neat the origin and becomes a constant equal to V_(0) for |x|impliesX_(0) (see figure). . The acceleration of this partile for |x|gtX_(0) is (a) proprtional to V_(0) (b) proportional to.

In a conservative force field we can find the radial component of force from the potential energy function by using F = -(dU)/(dr) . Here, a positive force means repulsion and a negative force means attraction. From the given potential energy function U(r ) we can find the equilibrium position where force is zero. We can also find the ionisation energy which is the work done to move the particle from a certain position to infinity. Let us consider a case in which a particle is bound to a certain point at a distance r from the centre of the force. The potential energy of the particle is : U(r )=(A)/(r^(2))-(B)/(r ) where r is the distance from the centre of the force and A andB are positive constants. Answer the following questions. If the total energy of the particle is E=-(3B^(2))/(16A) , and it is known that the motion is radial only then the velocity is zero at

In a conservative force field we can find the radial component of force from the potential energy function by using F = -(dU)/(dr) . Here, a positive force means repulsion and a negative force means attraction. From the given potential energy function U(r ) we can find the equilibrium position where force is zero. We can also find the ionisation energy which is the work done to move the particle from a certain position to infinity. Let us consider a case in which a particle is bound to a certain point at a distance r from the centre of the force. The potential energy of the particle is : U(r )=(A)/(r^(2))-(B)/(r ) where r is the distance from the centre of the force and A andB are positive constants. Answer the following questions. If the total energy of the particle is E=-(3B^(2))/(16A) , and it is known that the motion is radial only then the velocity is zero at