The document discusses error propagation in physics measurements and calculations. It provides formulas for calculating the uncertainty (error) when adding, subtracting, multiplying and dividing measurements. The key points are:
1) When adding or subtracting measurements, the total uncertainty is the square root of the sum of the individual measurement uncertainties squared.
2) When multiplying or dividing measurements, the total uncertainty is calculated by adding the individual relative uncertainties squared.
3) For measurements raised to a power, the relative uncertainty is equal to the relative uncertainty of the measurement itself multiplied by the power.
Lesson 6: Polar, Cylindrical, and Spherical coordinatesMatthew Leingang
"The fact that space is three-dimensional is due to nature. The way we measure it is due to us." Cartesian coordinates are one familiar way to do that, but other coordinate systems exist which are more useful in other situations.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
We evaulate the impact of measurement error on three components of a forest planning model: initial inventory, first period harvest and silvicultural activities chosen. Calculated initial inventory was off by as much as 15%, and significantly different objective function values and activity schedule also resulted.
Lesson 6: Polar, Cylindrical, and Spherical coordinatesMatthew Leingang
"The fact that space is three-dimensional is due to nature. The way we measure it is due to us." Cartesian coordinates are one familiar way to do that, but other coordinate systems exist which are more useful in other situations.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
We evaulate the impact of measurement error on three components of a forest planning model: initial inventory, first period harvest and silvicultural activities chosen. Calculated initial inventory was off by as much as 15%, and significantly different objective function values and activity schedule also resulted.
I split the presentation for the unit into two, as I added so many slides to help with student questions and misconceptions. This one focuses on mathematical aspects of the unit.
Types of errors
Among the most frequent sources of errors Brown counts
(1) interlingual transfer,
(2) intralingual transfer,
(3) context of learning,
and (4) various communication strategies the learners use
This presentation gives the information about Screw thread measurements and Gear measurement of the subject: Mechanical measurement and Metrology (10ME32/42) of VTU Syllabus covering unit-4.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
1. Error Propagation
The analysis of uncertainties (errors) in measurements and calculations is essential in the
physics laboratory. For example, suppose you measure the length of a long rod by
making three measurement x = xbest ± ∆x, y = ybest ± ∆y, and z = zbest ± ∆z. Each of these
measurements has its own uncertainty ∆x, ∆y, and ∆z respectively. What is the
uncertainty in the length of the rod L = x + y + z? When we add the measurements do the
uncertainties ∆x, ∆y, ∆z cancel, add, or remain the same? Likewise , suppose we
measure the dimensions b = bbest ± ∆b, h = hbest ± ∆h, and w = wbest ± ∆w of a block.
Again, each of these measurements has its own uncertainty ∆b, ∆h, and ∆w respectively.
What is the uncertainty in the volume of the block V = bhw? Do the uncertainties add,
cancel, or remain the same when we calculate the volume? In order for us to determine
what happens to the uncertainty (error) in the length of the rod or volume of the block we
must analyze how the error (uncertainty) propagates when we do the calculation. In
error analysis we refer to this as error propagation.
There is an error propagation formula that is used for calculating uncertainties when
adding or subtracting measurements with uncertainties and a different error propagation
formula for calculating uncertainties when multiplying or dividing measurements with
uncertainties. Let’s first look at the formula for adding or subtracting measurements with
uncertainties.
Adding or Subtracting Measurements with Uncertainties.
Suppose you make two measurements,
x = xbest ± ∆x
y = ybest ± ∆y
What is the uncertainty in the quantity q = x + y or q = x – y?
To obtain the uncertainty we will find the lowest and highest probable value of q = x + y.
Note that we would like to state q in the standard form of q = qbest ± ∆q where
qbest = xbest + ybest.
(highest probable value of q = x + y):
(xbest+ ∆x) + (ybest + ∆y) = (xbest+ ybest) + (∆x +∆y) = qbest + ∆q
(lowest probable value of q = x + y):
(xbest- ∆x) + (ybest - ∆y) = (xbest+ ybest) - (∆x +∆y) = qbest – ∆q
Thus, we that
∆q = ∆x + ∆y
is the uncertainty in q = x + y. A similar result applies if we needed to obtain the
uncertainty in the difference q = x – y. If we had added or subtracted more than two
1
2. measurements x, y, ......, z each with its own uncertainty ∆x, ∆y, ......... , ∆z respectively
, the result would be
∆q = ∆x + ∆y + ......... + ∆z
Now, if the uncertainties ∆x, ∆y, ........., ∆z are random and independent, the result is
∆q = (∆x) 2 + (∆y ) 2 + ......... + (∆z ) 2
Ex. x = 3.52 cm ± 0.05 cm
y = 2.35 cm ± 0.04 cm
Calculate q = x + y
We would like to state q in the standard form of q = qbest ± ∆q
xbest = 3.52cm, ∆x = 0.05cm
ybest = 2.35cm, ∆y = 0.04cm
qbest = xbest + ybest = 3.52cm + 2.35cm = 5.87cm
∆q = (∆x) 2 + (∆y ) 2 = (0.05)2 + (0.04) 2 = 0.06cm
q = 5.87cm ± 0.06cm
Multiplying or Dividing Measurements with Uncertainties
Suppose you make two measurements,
x = xbest ± ∆x
y = ybest ± ∆y
What is the uncertainty in the quantity q = xy or q = x/y?
To obtain the uncertainty we will find the highest and lowest probable value of q = xy.
The result will be the same if we consider q = x/y. Again we would like to state q in the
standard form of q = qbest ± ∆q where now qbest = xbest ybest.
(highest probable value of q = xy):
(xbest+ ∆x)(ybest + ∆y) = xbestybest + xbest ∆y +∆x ybest + ∆x ∆y = qbest + ∆q
= xbestybest + (xbest ∆y + ∆x ybest) = qbest + ∆q
(lowest probable value of q = xy):
2
3. (xbest- ∆x)(ybest - ∆y) = xbestybest - xbest ∆y - ∆x ybest + ∆x ∆y = qbest – ∆q
= xbestybest – (xbest ∆y + ∆x ybest) = qbest – ∆q
Since the uncertainties ∆x and ∆y are assumed to be small, then the product ∆x ∆y ≈ 0.
Thus, we see that ∆q = xbest ∆y +∆x ybest in either case. Dividing by xbestybest gives
∆q x ∆y y ∆x
= best + best
xbest y best xbest y best xbest y best
∆q ∆y ∆x
= +
qbest y best x best
Again, a similar result applies if we needed to obtain the uncertainty in the division of
q = x/y. If we had multiplied or divided more than two measurements x, y, ......, z each
with its own uncertainty ∆x, ∆y, ......... , ∆z respectively, the result would be
∆q ∆y ∆x ∆z
= + + ........ +
qbest y best x best z best
Now, if the uncertainties ∆x, ∆y, ........., ∆z are random and independent, the result is
2 2 2
∆q ∆y ∆x ∆z
=
y +
x + ........ +
z
qbest best best best
Ex. x = 49.52cm ± 0.08cm
y = 189.53cm ± 0.05cm
Calculate q = xy
We would like to state q in the standard form of q = qbest ± ∆q
xbest = 49.52cm, ∆x = 0.08cm
ybest = 189.53cm, ∆y = 0.05cm
qbest = xbestybest = (49.52cm)(189.53cm)=9.38553 x 103 cm2
2 2 2
∆q ∆x ∆y 0.08cm 0.05cm
= +
y
= 49.52cm + 189.53cm = 1.63691E − 3
qbest xbest best
3
4. ∆q = (1.63691 x 10-3)qbest = (1.63691 x 10-3) (9.38553 x 103 cm2)
∆q = 15.3632cm2 ≈ 20 cm2
q = 9390 cm2 ± 20 cm2
Uncertainty for a Quantity Raised to a Power
If a measurement x has uncertainty ∆x, then the uncertainty in q = xn, is given by the
expression
∆q ∆x
= n
qbest xbest
Ex. Let q = x3 where x = 5.75cm ± 0.08 cm.
Calculate the uncertainty ∆q in the quantity q.
We would like to state q in the standard form of q = qbest ± ∆q
n=3
∆x = 0.08cm
xbest = 5.75cm
qbest = xbest = 190.1cm 3
3
∆q ∆x
= n =
qbest xbest
∆q 0.08cm
3
= (3)
190.1cm 5.75cm
∆q = 7.93cm 3 ≈ 7cm 3
q = 190cm 3 ± 7cm 3
4