MECHANICAL FORMULAE BY RAMESH

work was first coined in the 1830s by the French mathematician Gaspard-Gustave Coriolis.[1]
According to the work-energy theorem if an external force acts upon an object, causing its kinetic energy to change from Ek1 to Ek2, then the mechanical work (W) is given by:[2]

where m is the mass of the object and v is the object's speed.
The mechanical work applied to an object can be calculated from the dot product of the applied force (F) and the displacement (d) of the object. This is given by:

Contents[hide]
1 Introduction
2 Units
3 Mathematical calculation
3.1 Force and displacement
3.2 Mechanical energy
4 Frame of reference
5 References
6 Bibliography
7 External links
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[edit] Introduction

A baseball pitcher does positive work on the ball by transferring energy into it. The catcher does negative work on it.
Work can be zero even when there is a force. The centripetal force in circular motion, for example, does zero work because the kinetic energy of the moving object doesn't change. Likewise when a book sits on a table, the table does no work on the book despite exerting a force equivalent to mg upwards, because no energy is transferred into or out of the book.
Heat conduction is not considered to be a form of work, since the energy gets transferred into atomic vibrations rather than a macroscopic displacement.

[edit] Units
Main article: work (thermodynamics)
The SI unit of work is the joule (J), which is defined as the work done by a force of one newton acting over a distance of one meter. This definition is based on Sadi Carnot's 1824 definition of work as "weight lifted through a height", which is based on the fact that early steam engines were principally used to lift buckets of water, through a gravitational height, out of flooded ore mines. The dimensionally equivalent newton-meter (N·m) is sometimes used instead; however, it is also sometimes reserved for torque to distinguish its units from work or energy.
Non-SI units of work include the erg, the foot-pound, the foot-poundal, and the liter-atmosphere.

[edit] Mathematical calculation

[edit] Force and displacement
Force and displacement are both vector quantities and they are combined using the dot product to evaluate the mechanical work, a scalar quantity:
(1)
where is the angle between the force and the displacement vector.
In order for this formula to be valid, the force and angle must remain constant. The object's path must always remain on a single, straight line, though it may change directions while moving along the line.
In situations where the force changes over time, or the path deviates from a straight line, equation (1) is not generally applicable although it is possible to divide the motion into small steps, such that the force and motion are well approximated as being constant for each step, and then to express the overall work as the sum over these steps.
The general definition of mechanical work is given by the following line integral:
(2)
where:

Theory and Design for Mechanical Measurements

However, the most serious problem with the book is the number of errors, some of such a fundamental nature that one is inevitably driven to consider just how much of the rest one can consider reliable.

Some instances will show what one must expect. These examples were not found by a careful trawl in depth, so must be considered as representative only.

The definition of the ohm as the resistance of a column of mercury was abandoned many years ago; a standard cell must never be used to deliver 100 mA, let alone for a few minutes, and in any case, both volt and ohm have been defined for more than ten years by quantum effects.

Problem 1.37 specifies an LVDT being driven by a d.c. voltage, when of course only a.c. can be used. (There are `d.c.' LVDTs but these contain electronics to generate the necessary a.c.) The accompanying diagram also shows the connections incorrectly and the text contains the sentence `outputs a voltage which is linear to the input'. Which input? The electrical one or the displacement? Such looseness abounds, even ignoring the use of `output' as a verb.

Page 36: `Analog describes a signal that is continuous in time'. Alas, not so, analog describes a signal where the intelligence is represented by a continuously variable level.

Page 43: Fig. 2.8 seems to show (quite clearly!) that subtracting a d.c. offset actually causes a waveform to change shape.

Page 197: `An a.c. meter indicates a true rms value for a simple periodic signal only'. Absolutely wrong: an a.c. meter (the rectified moving coil type is meant) indicates average only, whatever it may be scaled.

The authors' application of mathematical analysis to the consideration of errors is both praiseworthy and necessary but reveals their obvious preference for an analysis of errors rather than for the application of careful thought to remove sources of error first, before applying analysis to those that inevitably remain. An example of this lack of thought is revealed in Fig. 6.4, a simple multirange ammeter, where the moving coil would vaporize if the switch were ever operated under load or there were ever any jump in contact resistance. Amazingly there is no discussion of the universal shunt, nor of the necessity of a swamping series resistor.

There is very little discussion of what most people would actually think of when considering `Mechanical Measurement'. Importantly, there is no discussion on the fundamental and vital differences between a micrometer and a caliper - that the former indicates its own true zero, unless it is very seriously damaged, while the latter is subject to many errors caused by wear or by quite minor damage, a further indication of inattention to some vital considerations. The authors seem also to be unaware that micrometers can be used in at least as large a size as a caliper.

There is no mention at all of modern precision coordinate measuring machines - only one of several omissions.

The arrangement of the knife edges of a precision balance in Fig. 12.13, purporting to show a precision balance, is drastically wrong.

A pencil type pressure gauge has no spring, which rather makes nonsense of the rest of the analysis.

It is surprising and regrettable that so many errors (and there are many more) should have survived into a third edition. A major revision is called for before the book could be wholeheartedly recommended.

This book is certainly a useful addition to one's library for its chapters on probability of statistics and analysis on uncertainty, but it would be most unwise to use it for general guidance on the theory of measurements for one not already thoroughly familiar with the design of actual instruments.