Conservation of Mechanical energy

Introduction :
Conservation of mechanical energy states that the whole mechanical power (sum of kinetic energy+ potential energy) of a structure remains stable provided the forces acting on the body are conservative. The standard of conservation of mechanical energy states, the whole mechanical energy of a conservation system leftovers constant.

Proof of Conservation of Mechanical energy

Suppose a particle of mass m moves from position x₁ to another position x₂ under the action of conservation force. As a result its velocity changes from v₁ to v₂. By work-energy theorem, we have

If V₁ and V₂ are the potential energies at positions x₁ and x₂, we have

If effort done by a conservative force is positive, the kinetic energy of body increases and potential energy of body decrease to stay total mechanical energy conserved.

Example of Freely Falling Body

Gravitational force is conservation force. When a body of mass m, initially at rest at height H, above the ground, falls under gravity, then its total mechanical energy remains constant.

At heighest point A:
The body is at rest, therefore kinetic energy,

Potential energy, V=mgh
Total mechanical energy at A=K+V=0+mgH=mgH

At point B:
Let the body reach at intermediate point B at distance x below point.

Potential energy=mg(H-x)

At point C:
Let C be the point on the ground. If v_C is the velocity at point C, then relation
V²=u²+2gs gives v²_C =0 +2gH=2gH
Kinetic energy,

Potential energy = 0
Total mechanical energy at C=mgH=0=mgH
Thus it is clear that for a freely falling body, the total mechanical energy remains constant.

Example of Mass-Spring System in Conservation of Mechanical Energy

Consider a spring-block system located on a horizontal frictionless table. Mass of block is M and force stable of spring is K. when mass is in position A, spring is in its usual length. So that elastic PE of spring is zero. When mass is taken to position B, the spring is stretched by an amount x₀, so elastic potential energy of spring (= (1/2 )kx₀²) and kinetic energy in this position = zero.

Torque Equation of a D.C Motor

Introduction:
The turning or twisting force about an axis is called torque. Consider a wheel of radius R meters acted upon by a circumferential force F Newton’s .Basically the torque is developed in the armature and hence, gross torque produced is denoted as T_a. The mechanical power developed in the armature is transmitted to the load through the shaft of the motor.

Torque Equation of a D.C Motor

Let us see about examples of torque,
The wheel is rotating at a speed of N r.p.m
Then angular speed of the wheel is,



So, work done in one revolution is,
W  =   F * distance travelled in one revolution
      =   F * 2π R Joules
P   =   Power developed = Work done/ Time



P  = T *ω      Watts
T = Torque   in  N  - m
ω = Angular speed in rad / sec.

let T_a be the gross torque developed by the armature of the motor in examples of torque. It is also called armature torque. The gross mechanical power developed in the armature is E_b I_a, as seen from the power equation. So if speed of the motor is N r.p.m. then,
Power in armature = Armature torque * ω



But E_b in a motor is given by,


This is the examples torque equation of a d.c motor.

Example of torque

Let us see about examples of torque,
A 4 pole d.c motor takes a 50 A armature current. The armature has lap connected 480 conductors. The flux per pole is 20 mWb. Calculate the gross torque developed by the armature of the motor.
Solution:

Types of torque in the motor

In examples of torque,the mechanical power developed in the armature is transmitted to the load through the shaft of the motor. It is impossible to transmit the entire power developed by the armature to load.This is because while transmitting the power through the shaft, there is a power loss due to the friction, windage and the iron loss. the torque required to overcome these losses is called lost torque.

Diffraction patterns

A method is described which assigns indices to a set of single-crystal reflections recorded by the rotation-oscillation technique using a fixed X-ray wavelength. It is assumed that the space group and approximate unit-cell parameters are known. The unknown crystal orientation is determined directly from the observed diffraction pattern of one or several oscillation data records. A local indexing procedure is described which tolerates large initial errors in the parameters controlling the diffraction pattern. These parameters are refined subsequently, thereby satisfying the constraints imposed by the space-group symmetry.

Diffraction bending of waves around the edge of an obstacle. When light strikes an opaque body, for instance, a shadow forms on the side of the body that is shielded from the light source. Ordinarily light travels in straight lines through a uniform, transparent medium, but those light waves that just pass the edges of the opaque body are bent, or deflected. This diffraction produces a fuzzy border region between the shadow area and the lighted area. Upon close examination it can be seen that this border region is actually a series of alternate dark and light lines extending both slightly into the shadow area and slightly into the lighted area. If the observer looks for these patterns, he will find that they are not always sharp. However a sharp pattern can be produced if a single, distant light source, or a point light source, is used to cast a shadow behind an opaque body.
The radiation diffraction pattern for amorphous materials requires special analysis for information on atomic groupings. Powdered crystalline material may also give diffuse patterns requiring this type of analysis. Absence of a sharp diffraction pattern does not necessarily indicate absence of the material. Examples are given for chrysotile and powdered chrysotile.

If the diffraction pattern of the hyperbolic umbilical diffraction catastrophe is produced by an optical system of increasing aperture, it passes continuously from the two-dimensional system of Airy rings in the focal plane, made by a very small aperture, to the full three-dimensional pattern corresponding to infinite aperture. The paper studies this transition by examining the truncated diffraction integral and following the evolution of the wave dislocation lines (phase singularities) on which the pattern is based. The seed of the evolution from a two-dimensional to a three-dimensional pattern turns out to be already present asymptotically even for the smallest aperture: namely, a column of small dislocation rings very close to the axis that stream in procession towards the focal plane, and become dislocations lying in the Airy fringe surfaces that run parallel to the main fold caustic, only to disappear ultimately by retreat to infinity. The evolution into the final dislocation pattern takes place by a sequence of primitive local topological events, such as reconnection (hyperbolic interchange) and ring creation.

X Ray diffraction Analysis

Introduction
After the discovery of X-rays, scientist started working on the wave nature of these rays. To test the nature, X-rays have to produce interference and diffraction patterns. For diffraction to occur, the wavelength should be in the order of its slit width. But we know that X-ray have shorter wavelength and it is quite impossible to make such a slit / grating of smaller dimension. So the wave nature was studied in atomic level. Scientist by name Laue suggested that crystal can act like a space grating to observe diffraction. This experiment was later supported by Bragg’s equation.

X Ray diffraction Analysis : Bragg’s law

Consider a plane lattice crystal with inter planar distance d. Suppose a beam of X-rays of wavelength λ  is incident on the crystal at an angle θ , the beam will be reflected in all possible atomic planes. The path difference between any two reflected waves is equal to the integral multiple of wavelength. The ray P gets reflected from the surface while the ray Q has to under go some path difference. The extra distance traveled by the ray Q from the figure is
 (BC +CD). From the diagram either BC or CD is equal to d sin theta. So the path difference is
    d sin θ  + d sin θ  = n λ
    2 d sin θ  = n λ
 Here n is the order = 1,2, 3 …… .This is Braggs law

X Ray diffraction : Analysis of the Pattern

The crystal which is considered as the slit is placed in the Bragg spectrometer for investigation. X-rays are incident on the crystal at different angles and its corresponding ionization current is noted. The below is the plot of ionization current and incident angle
 
  At a certain values of the angle of incidence, the ionization current is increases abruptly or at peak value. Basing on the angle at maximum current, the planar distance can be calculated using Bragg’s law.

 When the angle increases i.e. order of the spectrum, the intensity of the X- rays decreases

Ionization current will never fall to zero.

Thus we can say the diffraction analysis of X-rays helped us to study the crystal structure.

Multiple slit diffraction

Introduction to diffraction:
                The wave nature of the light is further confirmed by the phenomenon of diffraction. The word diffraction is derived from the Latin word diffracts which means break to pieces. When the waves are encountering obstacles they bend round the edges of the obstacles if the dimensions of the obstacles are comparable to the wavelength of the waves. The bending of waves around the edges of an obstacle is called diffraction.

Diffraction

Diffraction in opening:
                The diffraction may be take place in single or multiple slits. Here we are going to see about the diffraction by considering the passage of waves through the opening. When the opening is large compared to the wave length the waves do not bend round the edges which is given as,
                                                                                 
             When the opening is small, the bending effect round the edges is noticeable. When the opening is very small the waves spread over the surface behind the opening. The opening appears to act as an independent source of waves which propagate in all direction behind the opening.

Multiple slit diffraction

Diffraction in multiple slit:
                  Here we are going to discuss about the diffraction at multiple slits. We have seen the narrow slit gives a diffraction pattern considering of a principal maximum flanked by secondary maxima of lower intensity. In case of the multiple slit, each slit produces the similar diffraction effects in the same direction and the observed pattern is crossed by a number of interferences fringes. The regions of first order, second order, etc. maxima contain equally spaced fringes but they will be progressively fainter.
                 The envelope of the intensity variation of the interference fringes is identical to that of the diffraction pattern due to a single slit. In general I_S is the intensity at a point due to interference of light from multiple slits and I_d is the intensity due to diffraction of a single slit, then the resultant intensity I is given by,
                                                                                    I = I_d `xx` I_S
                Hence if I_d=0 at any point, then I=0 at that point irrespective of the values of I_S. The intensity and sharpness of the principal maxima increase and those of the secondary maxima decrease.
                When the slits are large in number, bright narrow lines are visible on a dark background . The greater the number of slits and the closer they are the narrower and brighter are the lines on the screen. Bringing the slits closer results, of course, in an increase in the distance between the lines on the screen.

Laser diffraction

The first laser diffraction system designed by Malvern Instruments was first introduced in the 1970’s. Since then, the technique has been accepted across a wide range of applications as a means of obtaining rapid, robust particle size data. The pages below provide an introduction to how the technique works and how the results obtained compare to other methods of particle size analysis.

Tri-Laser Diffraction Technolology

The TRI-LASER Diffraction System developed by MICROTRAC allows light scattering measurements to be made from the forward low angle region to almost the entire angular spectrum (approximately zero to 160 degrees). It does so by a combination of three lasers and two detector arrays, all in fixed positions. The primary laser (onaxis) produces scatter from nearly on-axis to about 60 degrees, detected by a forward array and a high-angle array, both of which have logarithmic spacing of the detector segments. The second laser (off-axis) is positioned to produce scatter beyond the 60 degree level which is detected using the same detector arrays. The third laser (off-axis) is positioned to produce backscatter, again using the same detector arrays. This technique effectively multiplies the number of sensors that are available for detection of scattered light.

During a measurement cycle, Laser 1 is switched on while Lasers 2 and 3 remain inactivated. The sample to be measured scatters light in an angular pattern depending on the material size. The scattered light from Laser 1 is detected by the on axis, forward detector and the off axis, high angle detector. Laser 1 is then switched off and Laser 2 is activated. Laser 2 is directed at the sample at a different angle of incidence providing a different optical axis. Light scattered by the sample is detected by the same fixed detectors. Laser 2 is then switched off and Laser 3 is activated. Again the angle of incidence and optical axis is different. In this case the fixed detectors detect light that is back–scattered by the sample. The resultant scattered light information from all three lasers is combined to generate particle size distributions with unsurpassed resolution. Tri laser diffraction technology is proprietary and is patented by Microtrac.

Electron Diffraction

Electyron Diffraction refers to the wave nature of electrons. However, a from of technical or practical point of view, it may be regarded as a technique used to study matter by firing electrons at a sample and observing the resulting interference pattern. This phenomenon is commonly known as the wave particle duality, in which it states that the behavior of a particle of matter (in this case the incident electron) can be described by a wave. For this reason an electron can be regarded as a wave, like sound or water waves. This technique is similar to X-ray and neutron diffraction.

Electron diffraction is most frequently used in the solid state physics and chemistry to study the crystal structure of solids. Experiments are usually performed in the transmission electron microscope (TEM), or a scanning electron microscope (SEM) as electron back scatter diffraction. In these instruments electrons are accelerated by electrostatic potential in order to gain the desired energy and determine their wavelength before they interact with the sample to be studied.

Electron Diffraction in Materials Science

Electron diffraction is an very important technique for crystallographic characterization, a valuable complementary tool to powder and single crystal X-ray diffraction.

Applications include phase identification and precision determination of suitable structural details for crystals in the micrometer to nanometer size range.

Electron Diffraction with the PDF-4+ Database

• The PDF-4+ database can be used to generate two types of electron diffraction patterns for all PDF entries with atomic coordinates:
  1.   Transmission electron spot patterns
  2.   Electron back scattering patterns
• To illustrate electron diffraction pattern generation with the PDF-4+ database, Iron (Fe, face centered
• cubic (FCC), space group Fm-3m) will be used as an example.
• The PDF-4+ database cannot perform search-match procedures directly on digital electron diffraction patterns, they must first be indexed to obtain a d-spacing-intensity (d/I) list. Search-match procedures can then be performed using SIeve+.
• Electron diffraction patterns generated by the PDF-4+ database do not account for intensity variation due to either sample or instrumental effects.