Row reduction method and rank of a matrix

1. Matrices are just a display of set of numbers and it does not have any value For example
is a 2 by 3 matrix having 2 rows and 3 columns.
Aij represents a matrix element of i’th row and jth column for example here A12=6 and A21=-2
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Objective type questions(Electrostatics)

Question 1
Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then:
(a) negative and distributed uniformly over the sphere
(b) negative and appears only at the point on the sphere closest to the point-charge
(c) negative and distributed non-uniformly over the entire surface of the sphere
(d) zero

Question 2
Two similar point charges are situated at x-axis at point x=-a and x=+a . another point charge Q is placed at the origin. If Q is displaced by a small distance x on the X-axis , the change in its electric potential energy is approximately proportional to
(a) x
(b) x2
(c) x3
(d) 1/x

Question 3
At a point inside a charged hollow metallic sphere
(a) the potential is zero
(b) the electric field is zero
(c) the potential depends on the distance of the point from the center
(d) the electric field depends on the distance of the point from the center

Question 4
A hollow sphere of metal of radius 10cm is charged in such a way that the potential of its surface is 5 Volt. The potential at the center of the sphere is
(a) 0
(b) 5 volt
(c) 50 Volt
(d) equal to potential at a distance 10cm from the surface of the sphere

Question 5
A sphere of radius 2cm has a charge of 2 micro Coulomb while sphere of radius 5cm has charge 5 micro coulomb. The ratio of electric fields at a distance 10cm from the center of the spheres will be
(a) 2:5
(b) 1:1
(c) 5:2
(d) 4:25

Answer:
1. d
2. b
3. b
4. b
5. a

Question on fluid mechanics


Question :
A large open top container of negligible mass and uniform area of cross section A has a small hole of cross-sectional area A/100 in its side wall near the bottom. The container is kept on the smooth horizontal floor and contains a liquid of density and mass m0. Assuming that the liquid starts flowing out horizontally through the hole at t=0 , calculate
(i) acceleration of the container, and
(ii) its velocity
when 75% of the liquid has drained out

Solution:
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Rolling Friction

  • Consider a situation of the ring or a sphere rolling without slipping over a horizontal plane.In this case there is only one point of contact between the body and the plane

  • The frictional forces developed between two surfaces in case described above is called rolling friction

  • Rolling friction developes between two surfaces when one body rolls over the surface of another body

  • We know that it is very difficult to pull a heavy metal box on a rough surface and if we attah four metal wheel to the box it becomes easiar to move the box on the same surface

  • Thus resistance offered by the surface during rolling is relatively less than offered during sliding friction

  • This is because while rolling surfaces in contact do not rub each other

  • Rolling friction is negligible in comparision to the kinetic and static friction which are present simlutanously

  • In many parts of the machine where this type of friction is undesirable ball bearings(small steel balls) are generally kept between the rotating parts of the machines.This ways power dissipation during the motion can be reduced

Rigid Body Rotation Question 2


Question :
A carpet of mass M made of inextensible material is rolled along its length in the form of a cylinder of radius R and is kept on a rough floor. The carpet starts unrolling without sliding on the floor when a negligible small push is given to it. Calculate the horizontal velocity of the axis of the cylinder part of the carpet when its radius is reduced to R/2.

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How to apply law of conservation of energy in mechanics

We often comes across the problems in mechanics where we need to apply the law of conservation of energy where gravitational potential energy or gravity is involved . For solving such problems you can consider the following problem solving strategy,

  1. First of all define the system which includes all the interacting bodies . Now choose a zero point for gravitational potential energy according to your convenience.
  2. Select the body of interest and identify the point about which information is given in the question. Also identify the point where you want to find out asked quantity about the body of interest.
  3. Check for the possibility of the presence of non-conservative forces. If there are no non-conservative forces present then write down the energy conservation equation for the system and identify the unknown quantity asked in the question.
  4. Solve the equation for the unknown quantities asked in the question by substituting the given quantities in the equation obtained.

Mechanics :- Work and Energy (points to remember)


1. Work done on an object by a constant force F is
W=Fx
where F is the magnitude of the force, Δx is the magnitude of the displacement,and F and Δx point in the same direction.
SI unit of work : joule ( J) newton.meter
2. Work done is a simple number that is it is a scalar quantity not a vector. So there is no direction associated with it. Energy and energy transfer are also scalar quantities.
3. The kinetic energy KE of an object of mass m moving with a speed v is
defined by
T=1/2(mv2)
SI unit: joule ( J) kg.m2/s2
4. The net work done on an object is equal to the change in the object’s
kinetic energy: Wtot=Tf-Ti=ΔT
where the change in the kinetic energy is due entirely to the object’s change in speed
5. A force is conservative if the work it does moving an object between two points is the same no matter what path is taken.
6. The gravitational potential energy of a system consisting of the Earth and an object of mass m near the Earth’s surface is given by
V=mgy
where g is the acceleration due to gravity and y is the vertical position of the mass with respect to the surface of the Earth (or some other reference point).

Paramagnetic Substances

Paramagnetic substances are those materials which when placed in magnetic field becomes weakly magnetized in the direction of the external field. Some examples of paramagnetic substances are platinum, aluminium, chromium, manganese, copper sulphate, liquid oxygen etc. When a paramagnetic bar is placed in the magnetic field , the magnetic flux density in it is greater than the magnetic flux density B0 in the vacuum. Thus the relative permeability of the paramagnetic substances is slightly greater then 1. The magnetic flux density due to magnetization is small but positive. The susceptibility of these substances decreases with increase in temperature. Paramagnetic substances are used in the measurement of low temperature.
Properties:-

  • A paramagnetic bar , suspended between the poles of a magnet shows opposite poles to those of the magnet in its end.
  • Paramagnetic substances have a tendency to move from weaker to the stronger parts in a non-uniform magnetic field.
  • Paramagnetic substances obeys Curie's law  

Intensity of magnetization


  • Intensity of magnetization is denoted by I
  • It represents the extent to which the material is magnetized
  • When we place a material in the magnetic field , atomic dipoles of the material tends to align fully or partially in the direction of the field.
  • So, the net magnetic moment is developed in the direction of the field in any small volume of the material.
  • Intensity of magnetization is defined as the magnetic moment per unit volume of the magnetized material. So, I=M/V , where M is the total magnetic moment within volume V due to the magnetizing field i.e., M=∑ m
  • Unit of I is Am-1

Free electron model of atom and energy bands in solids


  • In atoms electrons orbits round the nucleus in their respective stable orbits.
  • Coulomb force due to nucleus on outermost electrons known as valence electrons is negligible.
  • These valence electrons are not bound with any particular atom and they are free to bind with any other atom in the crystal lattice.
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Superconductivity fact file


  • Bulk superconductor in a week magnetic field will act as a perfect diamagnet , with zero magnetic induction in the interior.
  • Nonmagnetic impurities have no marked effect on the SC transition temperature.
  • A sufficiently strong magnetic field will destroy SC. At critical temperature critical field is zero HC(TC)=0
  • Values of HC are always low for type I superconductors.
  • For a given HC the area under magnetization curve is same for type II SC as for type I SC.
  • In all SC entropy decreases markedly on cooling below transition temperature.
  • Superconducting state is the more ordered state.
  • Contribution to the heat capacity in the SC state is an exponential form with an argument proportional to -1/T
  • In SC the important interaction is electron-electron interaction which orders the electrons in K space with respect to the fermi gas of the electrons.
  • The argument of the exponential factor in the electronic heat capacity of a SC is found to be -Eg/2kT
  • The transition in zero magnetic field from the superconducting state to the normal state is the second order phase transition, not involving any latent heat but discontinuity in heat capacity.
  • Energy gap decreases continuity to zero as the temperature is increased to transition temperature.
  • For photons of energy less than energy gap , the resistivity of the superconductor vanishes at absolute zero.
  • As the temperature is increased not only does the gap decreases , but the resistivity for photon with energy below the energy gap no longer vanishes except at zero frequency.

Comparison between insulators and conductors

(1) Insulators
  • Insulators have very wide forbidden energy gap nearly of the order of 5eV or more.
  • Because of this very high energy gap it becomes impossible for electrons present in valence band to cross the gap and reach to the conduction band and this makes electrical conduction a practical impossibility in insulators at room temperature.
  • However at very high temperatures or with very high voltage applied across the ends of the insulator , it may conduct and this is termed as breakdown of an insulator.
(2) Conductors

  • Conduction band and valence band overlaps in case of a conductor.
  • Value of forbidden energy gap is zero for conductors in other words it does not exists at all.
  • For conductors or metals , valence band energies are same as conduction band energies and an valence electron can very easily become conduction electron (or, free electron) without any supply of heat energy.
  • This is why metals contain large number of free electrons even at room temperature and are good conductor of electricity.

Continuous spectrum and characteristic X-Rays?

When energetic electrons bombard atoms in a metal target (for ex. tungsten) an electron may be ejected from innermost K-shell, the atom then is in exited state and is unstable. If an electron from L-shell now moves to vacancy in K-shell , the energy of atom is decreased and simultaneously there is emission of radiation. If E is the change in energy when electron moves from L-shell to K-shell then ,
E=hν where h is the plank's constant and ν is the frequency of radiation.
Thus , ν=E/h and for high energies , wavelength of the radiation is short and is of the order of 10-8cm for X-Rays.
When we study X-Rays from a target it is observed to be a continuous spectrum with intense lines. These intense lines depends on the metal used as target and these are called characteristic X-Rays. The continuous spectrum depends on applied potential difference , current flowing in the filament and atomic number of target.

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