Motion And Its Laws

 Newton proposed three laws of motion:

i. Newton's first law of motion:
Everybody in the universe will remain in the state of rest or of uniform motion in a straight line unless no external force act on it. It gives the defination of force as "force is that external agent which changes or tends to change the state of rest or of uniform motion in a straight line".
Inertia is the tendency of a body to remain in its own state unless external force act on it. It is divided as:
a) Inertia of rest
b) Inertia of motion and
c) Inertia of direction
     Hence, Newton's first law of motion is also called Law of inertia.

 Momentum (P):
 The quantity of motion contained by a body which is equal to the product of mass and velocity is known as momentum.
   Therefore, Momentum(P) = m * v
  It is a Vector quantity.

ii. Newton's second law of motion:
It states that "the rate of change in momentum of a body is directly proportional to the force act on a body and displacement takes place in the direction of force".
    Therefore, force (F) = dp/dt
                                    = d(mv)/dt
                                    = mdv/dt + vdm/dt
    If m is constant then, F = ma
    If v is constant then, F = vdm/dt

iii. Newton's third law of motion:
It states that "for every action there is equal and opposite reaction". Action and reaction act on two different bodies so they never cancel eachother.

2. Impulse:
 The net effect of force acting on a body is measured by a quantity called impulse.
Hence, impulse (I) = P2 - P1 = Change in momentum.

3. Motion in a lift:
  When a person of mass (m) is standing on the floor of lift at rest R = mg is called weight of a person. For uniform motion of a lift a = 0 so,
 Net force = R - mg = ma
or, R - mg = 0
or, R = mg

 i. When lift is accelerating upward:
 When lift is accelerating upward with an acceleration (a) then,
R - mg = ma
or, R = mg+ma

 Reaction of floor measure the apparent weight of body which increases.

 ii. When lift is accelerating downwards:
  When lift is accelerating downwards with an acceleration (a) then,
mg - R = ma
or, R = mg - ma

 Reaction of floor decreases hence apparent weight of person body decreases.

4. Principle of conservation of linear momentum:
   When no external force act on the system of colliding bodies then the total linear momentum of system remains constant (conserved).
 Therefore, F = dp/dt
If F = 0 then, dp/dt = 0
hence, P = constant
or, mv = constant.
  If  two bodies of mass m1 and m2 moving with u1 and u2 collide then their velocities changes to v1 and v2 respectively then,
 m1u1 + m2u2 = m1v1 + m2v2
When two bodies move together after collision then,
 v = (m1u1 + m2u2) / (m1 + m2)


5. Free body diagram:
 While solving the problems relating the newton's law of motion, free body diagram is used. During free body diagram,only the body of our consideration is taken an all the process acting on it are drawn. The net force acting on a body gives the acceleration so, F1 = ma.


6. Rocket propulsion:
 The propulsion of rocket is based on the principle of conservation of linear momentum or newton's third law of motion. A rocket of initial mass M1 eject combust fuel at the rate of dm/dt at any instant at which M be the mass of rocket the velocity of ejected gas relative rocket is v then,
  i. Force on a rocket due to ejection of gas in absence of gravity is, F = dm/dt * v
  ii. Full acceleration on rocket due to this force is, a = F/M = dm/dt * v/M
  iii. Net force on rocket due to gravity, F = dm/dt * v - mg
  iv. Acceleration on rocket in gravity is, a = F/m = dm/dt * v/m - g

X-Rays

X-Rays were discovered by Roentzen in 1895. X-Rays are electromagnetic waves of short wavelengths in the range of 10Å to 0.5Å. They are produced when electrons with high speed strike a metal target of high atomic weight. The longer wavelength end of the spectrum is known as “soft x-rays” and the shorter wavelength end is known as “hard x-rays”. X-Rays are also called as Roentzen rays. All of them are not of single wavelength. Different X-rays have different wavelength but lines within the range.

Soft X-Rays:
a. Have long wavelength
b. low energetic and low penetrating power
c. wavelength above 4Å.

Hard X-Rays:
a. having short wavelength
b. more energetic and high penetrating power
wavelength is below 4Å.

 X-Rays Production:

X-rays are produced when fast moving electrons are suddenly stopped by a solid target. A coolidge tube is shown in the figure below:

The tube is exhausted to the best possible vacuum of the order of 10^-5 mm of mercury. The cathode consists of a tungsten filament (F) heated by a low tension battery. Thermoionic electrons emitted by a filament are accelerated towards a target (T) by a high P.D maintained between F and T. The filament is placed inside a metal cup G to focus the electrons on to the target. The target must be cooled to remove the heat generated in it by continuous electron-bombardment. The usual method is to mount the target material on a hollow copper tube through which cold water is continuously circulated. The target is made of a metal like tungsten or molybdenum having a high melting point and a high atomic number. Metals with high atomic number give more energetic and intense X-rays when used as targets.
In the time of X-rays production, majority percentage of incident power is converted into heat i.e more than 99%. Minority percentage of incident power is converted inro X-rays radiations i.e less than 1%.
In the Coolidge tube, the intensity and frequency of X-rays can be easily controlled.
The intensity of X-rays depends on the number of electrons striking the target per second. The number of electrons given out by the filament is proportional to its temperature, which can be adjusted by varying the current in the filament circuit. Therefore, the intensity of X-rays varies with the filament current.
The frequency of X-rays emitted depends on the voltage between the cathode and the anode(target). Let V be the accelerating potential across the tube, e be the charge on the electron then workdone on the electron in moving from the cathode to the anticathode= eV. The electron thus acquires Kinetic energy (K.E) which is converted into X-rays, when the electron strikes the target. If v(max) is the maximum frequency of the X-rays produced, then hv(max)= eV.
Therefore, the minimum wavelength produced by an X-ray tube= hc/eV.

Properties of X-rays:
a. X-rays are elecromagnetic waves.
b. X-rays are not deflected by fields (electric and magnetic).
c. They move along the straight path with the velocity of light in vacuum.
d. X-rays can cause photoelectric effect.
e. X-rays undergoes reflection, refraction, diffraction, interference and polarization.
f. They produce illumination of fluorescent materials on which they fall.
g. They ionize the gas through which they pass.
h. They can penetrate thin materials like wood, thin sheet etc.
i.  They donot pass through heavy metals and bones
j.  X-rays cast their shadow on the screen.
k. They affect photo-graphic plates.
l.  X-rays fall on the metal surface having high mass number, secondary X-rays are produced.

Semi-conductors

Semiconductor is a material which is neither a good conductor of electricity nor a good insulator. Simply, Those solid substance whose elecrical conductivity lies between good conductors and insulator are known as Semiconductors. Its conductivity lies midway between a conductor and an insulator. The resistivity of semiconductors varies from 10^-5 to 10^-4 ohm-m as compared to the values ranging from 10^-8 to 10^-6 ohm-m for conductor and 10^7 to 10^8 ohm-m for insulator.Examples of such substances are the crystalline forms of the fourth group of the periodic table. Germanium(Ge) and silicon(Si) are two very typical substances showing this behaviour.The band gap of semi-conductors varies from 0.2 to 2.5 eV which is quite small as compared to that of insulators. Eg: The band gap of diamond (a typical insulator) is 6 eV. The valence band and conduction bands of metals may even overlap.
At absolute zero temperature, a semiconductor behaves as an insulator because all the electrons are filled in its valence band and the conduction band is empty. But when the temperature increases the electrons starts to jump to the conduction band so that conductivity increases and resistivity decreases. Hence, the increase in temperature has negative coefficient of resistance in semi-conductor.

Currents in the Semiconductor:
There are two types of current in semiconductor which are classified as:
Electron current: The electric current which is set up in the semi-conductor due to the movement of free electrons in its conduction band is called electron current.
Hole current: The electric current which is set up in the semi-conductor due to the movement of the holes is called hole current. Its direction is opposite to the electron current.

Types of Semi-conductor:
There are two types of Semi-conductor:

Intrinsic semiconductor: The semiconductors like silicon (Si) and germanium (Ge) which are found in their pure state are called Intrinsic semiconductors or pure semiconducrors. In Intrinsic semiconductor the electric current is set up by thermally generated electrons and holes which is in very less amount and cannot be used significantly.
Now to increase the electrical conductivity of semiconductor other impurity atoms can be mixed with pure semiconductors. The process of mixing of impurity atoms with pure semiconductor atoms is called dopping and the impurity agent which is mixed is called dopping agent.

Extrinsic semiconductor: The semiconductors which are obtained by mixing with other impurity atoms in suitable amount with pure semiconductors are called Extrinsic semiconductors. Generally, one impurity atom is mixed with 8 pure atoms. Now the electrical conductivity of the semiconductor can be highly increased. On the basis of mixing of impurity atoms there are two types of extrinsic semiconductors which are listed below:

N-type extrinsic semiconductor: The extrinsic semiconductor which is obtained by mixing pentavalent impurity atom like As, P and antimony (Sb) then it is called N-type extrinsic semiconductor. It is called N-type because the majority charge carriers of such semi-conductors are free electrons.
Donor atom: In N-type semiconductor the pentavalent impurity atoms which are mixed to the semiconductor atoms donate an electron for the electrical conductivity, so these are called donor atoms. When they donate an electron, they are positively charged.

P-type extrinsic semiconductor: The extrinsic semiconductor which are formed due to the mixing of trivalent impurity atoms like boron (B), aluminium (Al), galium (Ga) etc with pure semi-conductor atoms are called P-type extrinsic semiconductor. Here P stands for positive. It means the majority charge carriers of P-type semiconductors are holes and minurity charge carriers are free electrons. Only the fraction of total current is obtained from free electrons.
Acceptor atom: When trivalent impurity atom is mixed with pure semiconductor atom, one bond is incomplete and creates a hole. This hole has a tendency to attract the free electrons i.e, it accepts the electrons. So, trivalent impurity atoms are called acceptor atoms.

Why Semiconductor is damaged by the strong current?
Ans: When strong current is passed through the semiconductor, it heats up the atoms in the covalent bonds of semiconductor. Now the covalent bonds are broken and the electrical conductivity of semiconductor increases. It means it loses the property of semiconductor and shows the behave of conductor. Hence, the semiconductor is damaged by the strong current.

Optical Instruments and Photometry

1. Defects of vision:

a.The least distance upto which an object can be clearly seen by a naked eye is called the least distance of distinct vision which is 25cm for normal eye.
b.The farthest point for the clear vision is infinity.
c.Ability of eye less to change its focal length is call power of accomodation.
d.When eye is relaxed it has maximum focal length and minimum focal length when eye is most strained(25cm).
e.The limit of resolution of eye is one minute.
f.The 355 resistance of vision of human eye is (1/10) sec.

1.1. Myopia (short sightedness):
In it distant object are not clearly visible. Image of the objects from before the retina. This defect can be removed by using spectacles having divergent lens. Suppose a person can see an object at maximum (x) then to see the distant object a divergent lens has to be introduced which has a virtual image of the object at a distance of x from the eye.
   i.e. u = Infinity
     V = f = -x
     Therefore, power of the lens, P = 1/f = -(1/x)

1.2. Hypermetropia (longsightedness):
In it near objects are not clearly visible. Image of the objects form behind the retina. This deftect can be removed by using spectacles having convergent lens.
i.e. u = D
    V = -d
   Or, 1/f = 1/u + 1/V
Therefore, 1/f = 1/D - 1/d.

1.3 Presbyopia:
In this defect both near and far objects are not clearly visible. This defect takes place at old age and is called old age defect. This defect is remedied by using bifocal lens. It is due to the loss in elasticity of ciliary muscles.

1.4. Astigmatism:
It is not equally clear in two mutually perpendicular directions which is due to the uneven curvature of the cornea. This defect is corrected by using a cylindrical lens.
     Visual angle: It is the angle subtended by an object at the eye . It is maximum when the object is at the least distance of distinct vision.
     i.e. visual angle = h/D

2. Microscope:
It is an optical instrument used to increase the visual angleof near objects which are too small to be seen by our naked eye. Microscope are of two types viz, simple microscope and compound microscope.

2.1. Simple Microscope:
It is also known as magnifying glass or magnifier and consists of a convex lens with object between its focus and optical centre. The image formed by it is errect, virtual, enlarged and on the same side of lens.

2.2. Compound Microscope:
It consists of two convex lens of short focal length, objective lens and eye piece. Object is outside the focus of objective which forms real image and acts as object for the eye piece. Depending on the adjustment, the image can be formed at the least distance of distinct vision or at the infinity.

3. Telescope:
It is an optical instrument used to increase the visual angle of distant objects. They are of three types:
3.1. Astronomical telescope: It consists of two convex lens, objective lens of large focal length and aperture and eye piece of small focal length and aperture. Object is at the infinity, so the image is formed at the focus of the objective lens which acts as the object for eye piece.

3.2. Terrestrial telescope: It is used to see distant object on the earth. The final image is virtual, errect and diminished.

3.3. Galilean Telescope: It is also a type of terrestrial telescope but of much smaller field of view. It's objective lens is a convergent lens while the eye piece lens is divergent lens. The final image is virtual, errect and diminished.

4. Photometry: 
The branch of optics which study and measure the light emitting capacity of a source and illuminance produced by it.

Radiant Flux (R):
The total energy radiated by a source per second is known as Radiant Flux (R). It's unit is watt.

Luminous Flux (Ø):
The light energy radiated by a source in one second is called Luminous Flux (Ø). It's unit is lumen.

Luminous intensity (I):
The luminous flux per unit solid angle is known as Luminous intensity (I). It's unit is candela i.e, lumen per steradian.

Illuminance (E):
The luminous flux per unit area falling normally is known as Illuminance.
It's unit is lumen/m² or lux.
Therefore, E = Ø/A = I/r² (For point source)
E = Ø/A = (4(pie)I) / 2(pie)rl = 2I/rl
Therefore, E is directly porportional to 1/r ( For a cylindrical source)

Lambert Cosine Law:
It states that " at a given point for a given source illuminance varies linearly with cosine of angle of incidence ". i.e E is directly porportional to cos(theta).

Photometer: 
It is a device use to compare illuminating power of two sources.
Two sources placed at a distance r1 and r2 form a screen having same illuminance then,
E1 = E2
or, I1/r1² = I2/r2²
or, I1 /I2 = ( r1 /r2 )²

Useful Units, Dimensions And Error Analysis

1.1 Physical Quantities:
     Different quantities needs to describe the physical phenomenon or object are called physical phenomenon or object are called physical quantities. Examples: speed, density, mass, length etc.
     The physical quantities are divided into two groups:

1. Fundamental Quantities: Those physical quantities which are independent to any other physical quantities are known as Fundamental quantities. Eg: mass, length, time etc.
2. Derived Quantities: Those physical quantities which depend on other physical quantities and only obtained by multiplying and dividing the fundamental quantities are known as Derived quantities. Eg: density, velocity, acceleration, force etc.

1.2 Units: 
     The physical quantities are measured by comparing with some standard measurement of same kinds are called units.
      The units are divided into two groups:
1. Fundamental Unit: Units of the fundamental quantities are called fundamental units. These are independent to any other units.
2. Derived Units: Units of the derived quantities are called derived unit. These units are obtained by multiplying and dividing the fundamental units.

   System of measurement:
i. CGS system: The system of measurement in which 3 fundamental quantities mass, length and time are measured in gm, cm and s respectively is known as CGS system. All other derived quantities are also measured in terms of these units of measurement.

ii. MKS system: The system of measurement in which 3 fundamental quantities mass, length and time are measured in kg, m and s respectively is known as MKS system. All other derived quantities are measured in terms of these unit of measurement.

iii. FPS system: The system of measurement in which 3 fundamental quantities mass, length and time are measured in pound (lb), foot (ft) and second (s) respectively is known as FPS system. All other derived quantities are measured in terms of these unit of measurement.

iv. International system of units (S.I) : S.I is an abbreviation " Le systeme International d " units which is french and equivalent of international system of unit. It is used widely throughout the World in which seven different quantities are introduced as fundamental quantities and their units as fundamental units.

     Quantity                       Unit                    Symbol
1.  Mass                           Kilogram              Kg
2.  Length                        Meter                    m
3.  Time                           Sec                        S
4.  Temperature               Kelvin                   K
5.  Electric Current          Ampere                 A
6.  Luminious Intensity   Candela                 cd
7.  Amount of                  Mole                     Mol.
     Substance

Two more quantities are introduced as supplementary quantities and their units as supplementary unit.

   Some useful practicle units:
1.Plane  angle                Radian                Rad
2.Solid angle                 Steradian             Sr

1.Astronomical unit (AU): Average distance between centre of Earth and centre of Sun.
     Therefore, 1AU = 1.496 * 10^11 = 1.5 * 10^11 m

2.Light year(ly): Distance travel by light in a vacuum in 1 year.
    Therefore, 1ly = 9.46 * 10^15 m

3.  1 inch = 2.54cm
     1 foot = 30.48cm
     1 yard = 91.44 cm
     1 mile = 1.609 * 10^3 m
     1 nautical mile = 1.852 * 10^3 m
     1 angstrom = 10^-10 m
     1 Fermi = 1femtometre = 10^-15
   
For Areas
    1 barn = 10^-28 m^2
    1 acre = 4047 m^2
    1 hectare  = 10^4 m^2
For mass
    1 tonne/metric ton = 1000 kg
    1 Quintal = 100 kg
    1 slug = 14.57 kg
    1 lb = 0.4536 kg
    1 amu = 1.67 * 10^-27 kg
For time
    1 shake = 10^-8 sec
    1 solar year = 365.25 days
For pressure
    1 bar = 1 atmospheric pressure = 10^5 N/m^2
    1 torr = 1 mm of Hg = 133N/m^2
    1 bar = 760 torr
    1 atmospheric pressure = 760 mm of Hg = 1.01 * 10^5 N/m^2

1.3 Dimension:
       The power of fundamental quantity involved in any physical quantity is called Dimension of that physical quantity. The representation of physical quantity interms of power of fundamental quantities involved in it is called dimensional formula of that physical quantity. Three fundamental quantities mass,length and time are represented by [M] , [L] and [T] respectively. All other derived quantities are also expressed in terms of these representations.
         E.g: Force = ma = Kgm/s^2 = [MLT^-2]
                Work = F * d = Nm(J) = [ML^2T^-2]
                Pressure = F/A = N/m^2 = [ML^-1T^-2]

1.4 Error analysis: (formula)
   Least count (LC): The smallest measurement that can be taken by an instrument is called Least count. E.g = Least count of scale graduates in mm is 1mm.
   Absolute error: The different between true value and measured value is called Absolute error.
       i. Error in sum of  quantities
       ii. Error in difference of quantities
       iii. Error in product of quantities
       iv. Error in division of quantities
       v. Error in quantities raised to some power.