Name: accuracy and approximation in calculations
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 28 s
Channel: JoVE

K12 Standards

Texas Essential Knowledge and Skills

High School

Specialised Topic in Science

112.51.c.1.

112.51.c.2.C.

Accuracy, limits, and approximation

בחישובים הנדסיים, דיוק, גבולות וקירוב משמשים לייצוג ערך.
שקול שני בלוקים מרובעים של ממדים שונים. ניתן לקבל את הערכים המדויקים של האזורים שלהם על ידי עיגול מספריהם לערכים הקרובים ביותר בהתבסס על נתונים משמעותיים.
חשבו על חרוט עגול ימני בעל רדיוס בסיס, r וגובה, h. אם הקונוס פרוס בצורה מעגלית במרחק x מהקודקוד שלו, כך שלאלמנט הפרוס יש עובי, Δx, אז ניתן לקבוע את נפחו.
על ידי החלת המגבלה מ- Δv ל- dv ו- Δx ל- dx, והזנחת הפרשים מסדר גבוה יותר, מתקבל הביטוי עבור אמצעי האחסון.
חשבו על קשת AB, תוך עדינות זווית קטנה ב-O. ניתן לקרב את אורך הקשת לבסיס של משולש ישר זווית.
אם אורך ההיפוטנוס הוא אחדות, אורך הקשת שווה לסינוס תטא, שווה בקירוב לתטה. בנוסף, קוסינוס תטא גם שווה בערך לאחדות.
עבור תטא של מעלה אחת, הערך המשוער עבור sin theta ו- tan theta הוא כמעט זהה.

<ol>
	<li>מרים, י"ל; Kraige, L.G., ובולטון, J.N. (2020). מכניקה הנדסית - סטטיקה. הובוקן, ניו ג'רזי: ג'ון ווילי. עמ' – 13-14.</li>
</ol>

דיוק, גבולות וקירוב

<div style="direction:rtl">דיוק, מגבלות וקירובים נפוצים בתחומים רבים, במיוחד בחישובים הנדסיים. מושגים אלו הכרחיים כדי להבטיח שערך נתון יהיה קרוב ככל האפשר לערכו האמיתי. דיוק מוגדר כקרבת הערך הנמדד לערך האמיתי או הממשי. במכניקה הנדסית, מדידות חוזרות ונשנות נלקחות במהלך ניתוחים תיאורטיים או ניסויים כדי להבטיח שהתוצאה מדויקת . הדיוק של כל פתרון מבוסס על הנתונים האמפיריים והתוצאות המתקבלות מחישובים או מדידות שונות. חשבו על מיכל גלילי מלא בנוזל. ניתן לקבוע את נפח הנוזל על ידי מדידת מידות המיכל וחישוב הנפח באמצעות נוסחה מתאימה. עם זאת, בפועל, אי אפשר למדוד את מידות המיכל בדיוק מוחלט. קירובים שימושיים במיוחד כאשר מתמודדים עם פונקציות מורכבות שקשה לפתור אותן בצורה אנליטית. מידת הקירוב מתייחסת למספר הדמויות המשמעותיות המשמשות במדידות. כאשר משתמשים בנתונים משמעותיים יותר, התוצאה מדויקת יותר. בעוד שקירובים שימושיים, יש להיזהר בעת השימוש בהם, מכיוון שהם עלולים להציג שגיאות המשפיעות על דיוק החישוב. משמעות הדבר היא שחיוני להבין את השגיאה שהביאה הקירוב וכיצד היא עשויה להשפיע על התוצאה. לדוגמה, עבור זווית קטנה יחסית, בדרך כלל פחות או שווה ל-°15 או 0.26 רדיאנים, ניתן להשתמש בקירוב כדי לפשט את פונקציות הסינוס, הקוסינוס והטנגנס. יש לציין שהדיוק של קירובים אלה יורד כאשר הזווית גדולה מ-°15 או 0.26 רדיאנים. עם זאת, עבור זוויות פחות או שווה ל-°15 (0.26 רדיאנים), לנוסחאות אלו יש דיוק טוב ומייצרות ערכים כמעט שווים לערך המדויק. המושג של גבולות חשוב לא פחות בחישובים הנדסיים. גבולות משמשים לבירור התנהגות של פונקציה, במיוחד כאשר משתנה בלתי תלוי מתקרב לערך מסוים. שקול פונקציה שמתקרבת לאינסוף כאשר x מתקרבת לערך מסוים. הגבול מגדיר את הערך המקסימלי שניתן להגיע אליו מבלי לעבור את הערך שצוין של x. מגבלות שימושיות במיוחד בקביעת היציבות והביצועים של מערכות ביישומים קריטיים.</div>

דיוק, גבולות וקירוב נפוצים בתחומים רבים, במיוחד בחישובים הנדסיים. מושגים אלה הם הכרחיים כדי להבטיח כי ...

Education

JoVE Core

Mechanical Engineering

הנדסת מכונות

Unit 1

יחידה 1

An Introduction to Statics

מבוא לסטטיקה

KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

accuracy and approximation in calculations

Accuracy and approximation are important in calculations, especially in engineering. Accuracy means how close a measured value is to the true or actual value. Repeated measurements are often used in engineering mechanics to help make results more precise and accurate.

The accuracy of a solution depends on empirical data and the results from different calculations or measurements. A cylindrical container filled with liquid is a simple example. Its volume can be found from the measured dimensions and a formula, but those dimensions cannot be measured with absolute precision in practice.

Approximations are useful when a function is too complex to solve exactly. The degree of approximation depends on the number of significant figures used in the measurements. More significant figures usually give a result that is closer to the true value.

Approximations can also introduce error, so the effect of the error must be understood. For small angles, usually less than or equal to 15° or 0.26 radians, sine, cosine, and tangent can be approximated with good accuracy. This accuracy becomes worse when the angle is greater than 15° or 0.26 radians.

Limits are another key idea in engineering calculations. A limit describes the behavior of a function as an independent variable approaches a specific value. For a function that grows toward infinity as x approaches a certain value, the limit helps define the maximum value that can be reached without passing that x value. Limits are useful for judging the stability and performance of systems in critical applications.

Accuracy and Approximation in Calculations

accuracy-and-approximation-in-calculations

an introduction to mechanics

Mechanics in Physics and Engineering

Mechanics is a key branch of physics that explains how objects stay at rest or move when forces act on them. It also has major uses in engineering and science. The topic helps students understand everyday motion, machine behavior, and the way many systems work. The recorded history of mechanics begins with Aristotle, who lived from 384 to 322 BC. He linked mechanics with physical theory and looked for a broad universal explanation. Later, Newton defined mechanics as the study of objects at...

basic concept

Fundamental Quantities in Engineering Mechanics

Engineering mechanics begins with four basic quantities: length, time, mass, and force. These ideas help describe how objects move and how they respond to loads. They also support many other engineering fields. Length measures the distance traveled by an object. Engineers describe this distance with coordinates relative to a reference point. Length is also useful for showing the geometry of physical systems. Time measures the succession of an event. A bus traveling from city X to city Y is a...

newton's law of motion

Newton's Laws and Everyday Motion

Newton's laws explain everyday motion and the forces that change it. These three laws describe why objects move, stay still, or speed up. They also show how force and mass affect motion. Newton's first law is the law of inertia. It says an object at rest stays at rest, and an object in motion keeps moving at a constant speed and direction unless an external force acts on it. A sliding box that slows down on its own is a good example. In that case, friction is the net external force that works...

newton's law of gravitational attraction

Gravity Between Masses and Distance

Newton’s law of gravitational attraction describes how gravity acts between two objects. Sir Isaac Newton built this law from empirical evidence and inductive reasoning. He published it in Philosophiae Naturalis Principia Mathematica, or the Principia, on July 5, 1687. The law says that the gravitational force between any two objects depends on their masses and their distance apart. The force is proportional to mass and inversely proportional to the square of the distance between the objects.

units of measurement

SI and Customary Measurement Systems

Mechanical engineering depends on units of measurement for precise and accurate calculations. These units give engineers a standard way to express and compare physical quantities. They are essential when designing and analyzing machines and mechanical systems. Many historical measurement systems have been used over time. Examples include the Babylonian system, the Roman system, the Egyptian system, the Olympian system, the British system, and the Indus Valley system. Today, the International...

international system of units

SI Units and Metric Prefixes

SI units and metric prefixes provide a standard way to measure size, mass, and time. The International System of Units, or SI system, is used worldwide because it gives scientists a common language for measurement. Its values do not change with location, which helps keep results consistent. The SI system is built on three base units: meter, kilogram, and second. These are the core units for length, mass, and time. The kilogram is the only base unit that already includes a prefix. Prefixes...

numerical calculations

Engineering Notation and Significant Figures

Numerical calculations in engineering depend on clear values, correct units, and careful rounding. Presenting the answer is an important part of engineering practice, so numerically accurate results are preferred. Handheld calculators and computers are often used because they help produce accurate answers. Problems can be solved in several ways. Solving algebraic symbols by hand is common, and the graphical method is often preferred in some cases. Computers are especially useful when many...

accuracy, limits, and approximation

Accuracy and approximation are important in calculations, especially in engineering. Accuracy means how close a measured value is to the true or actual value. Repeated measurements are often used in engineering mechanics to help make results more precise and accurate. The accuracy of a solution depends on empirical data and the results from different calculations or measurements. A cylindrical container filled with liquid is a simple example. Its volume can be found from the measured...

dimensional analysis

Using Dimensions to Check Physics Equations

Dimensional analysis is a useful way to check physics and engineering equations. It works by writing physical quantities in terms of basic dimensions such as mass, length, and time. Quantities like velocity, acceleration, and force are derived from these basic dimensions. This method helps compare measurements even when the units are different. It also checks whether an equation is dimensionally homogeneous. That means the powers of the fundamental dimensions must match on both sides of the...

problem solving in statics

Statics Problem-Solving Steps for Beams

Statics problem-solving uses a step-by-step method to analyze bodies in equilibrium, especially beams. It is an important part of physics and engineering because the goal is to find practical and accurate answers for real situations. The process usually starts with a physical situation and then turns it into a mathematical model. That model often needs approximations and assumptions. Very small distances can be ignored when they are much smaller than the main dimensions, such as beam width...

key terms and concepts

Learn how accuracy, significant figures, small-angle approximations, and limits help describe real measurements and function behavior in engineering calculations.

Accuracy and Approximation in Calculations | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Physics

PHYS.2

PHYS.2.A

Chemistry and Physics

IPC.2

IPC.2.A

The Scientific Method

The scientific method is a detailed, empirical problem-solving process used by biologists and other scientists. This iterative approach involves ...

Inductive Reasoning

Inductive reasoning is a form of logical thinking that uses related observations to arrive at a general conclusion. It is uncertain and operates in ...

Deductive Reasoning

Deductive reasoning, or deduction, is the type of logic used in hypothesis-based science. In deductive reasoning, the pattern of thinking moves in the ...

Correlation and Causation

Correlation and CausationStatistical tests can calculate whether there is a relationship, or correlation, between independent and dependent variables. A ...

Ask questions and define problems related to specialized topics of study based on observations or information from text, phenomena, models, or investigations;

112.51.c.1.A.

Scientific Method

The scientific method is used to solve problems and explain phenomena. The development of the scientific method coincided with changes in philosophy ...

Demonstrating the Scientific Method with Tree Branches
ExpandObtain branches containing a few dozen typical leaves from five different local tree species. ...

Observation and Background Research
ExpandNOTE: The first two steps in the scientific process involve studying and observing natural phenomena of ...

Problem-Solving

A problem-solving strategy is a plan of action used to find a solution. It involves trial and error, algorithms, heuristics, and insights. Effective ...

Problem Definition

Problem definition is the initial and the most essential step in research.
The researcher identifies and explains the specific issue or gap to be ...

Apply science practices related to specialized topics of study to plan and conduct investigations or use engineering practices to design solutions to problems;

112.51.c.1.B.

Proper Personal Protective Equipment

Robert M. Rioux &#38; William A Elliott, Pennsylvania State University, University Park, PA
Hazards are many and varied in the laboratory, but the right ...

Emergency Eyewash and Shower Stations

Robert M. Rioux, Pennsylvania State University, University Park, PA
The Occupational Safety and Health Administration (OSHA) mandates that an emergency ...

Electrical Safety

Robert M. Rioux &amp; Suprita Jharimune, Pennsylvania State University, University Park, PA
Among the many hazards present in the laboratory, electrical ...

Working with Centrifuges

Source: Robert M. Rioux &amp; Suprita Jharimune, Pennsylvania State University, University Park, PA
Earth's gravitational force is capable of separating ...

Working with Hot and Cold Sources

Source: Robert M. Rioux &amp; Suprita Jharimune, Pennsylvania State University, University Park, PA
Working with extreme temperatures, both high and low, ...

guidelines-in-case-of-a-laboratory-emergency

Guidelines in Case of a Laboratory Emergency

Robert M. Rioux &amp; Zhifeng Chen, Pennsylvania State University, University Park, PA
The most common laboratory emergencies include chemical spills, ...

chemical-storage-categories-hazards-and-compatibilities

Chemical Storage: Categories, Hazards And Compatibilities

Source: Robert M. Rioux &amp; Taslima A. Zaman, Pennsylvania State University, University Park, PA
While the use of various chemicals in experimental ...

Safe Handling of Mineral Acids

Source: Robert M. Rioux &#38; Taslima A. Zaman, Pennsylvania State University, University Park, PA
A mineral acid (or inorganic acid) is defined as a ...

Handling Chemical Spills

Source: Robert M. Rioux &#38; Taslima Zaman, Pennsylvania State University, University Park, PA
Since chemicals are commonly used for laboratory research ...

Proper Use of Autoclaves

Robert M. Rioux &amp; Zhifeng Chen, Pennsylvania State University, University Park, PA
Autoclaving is one of the most commonly used methods in the ...

Fume Hoods and Laminar Flow Cabinets

Robert M. Rioux &amp; William A. Elliott, Pennsylvania State University, University Park, PA
Fume hoods and laminar flow cabinets are engineering controls ...

Decontamination for Laboratory Biosafety

Robert M. Rioux and Zhifeng Chen, Pennsylvania State University, University Park, PA
Decontamination is essential for laboratory biosafety, as the ...

Proper Waste Disposal

Robert M. Rioux and Taslima A. Zaman, Pennsylvania State University, Pennsylvania, PA

Users are responsible for the proper disposal of the waste ...

Use appropriate safety equipment and practices during laboratory, classroom, and field investigations as outlined in Texas Education Agency-approved safety standards;

112.51.c.1.C.

Common Lab Glassware and Uses

Source: Laboratory of Dr. Neal Abrams &#8212; SUNY College of Environmental Science and Forestry
Glassware is a regular appearance in the professional ...

Lab Techniques

Measurement Accuracy
The ability to repeat an experiment and get the same results, or reproducibility, is essential in scientific research. However, ...

Source: Smaa Koraym at Johns Hopkins University, MD, USA

	Preparation of LaboratoryExpand
	Here we show the laboratory preparation for 10 students ...

Source: Smaa Koraym at Johns Hopkins University, MD, USA

	Determining the Density of an EggExpand
	Density is a measure of how compact a substance or ...

Proper Lab Notebook Keeping

Organic Lab Notebook Keeping
Record keeping is a vital skill in science and organic chemistry labs. Without proper notes, descriptions, quantities, or ...

Source: Lara Al Hariri at the University of Massachusetts Amherst, MA, USA

	Preparation of the LaboratoryExpand
	Here, we show the laboratory preparation ...

Source: Lara Al Hariri and Ahmed Basabrain at the University of Massachusetts Amherst, MA, USA

	Nitration of Methyl BenzoateExpand
	From this lab onward, ...

Use tools appropriate to the specialized topic of study;

112.51.c.1.D.

SI Units: 2019 Redefinition

Measurement is integral to analytical chemistry. Each record comprises a number &#8211; denoting the magnitude&#160; &#8211; and the unit &#8211; a ...

International System of Units

The International System of Units, known as the SI system, is a universally accepted measurement system recognized and used worldwide. The SI system is ...

Measurement: Standard Units

Every measurement provides three kinds of information: the size or magnitude of the measurement (a number), a standard of comparison for the measurement ...

Measurement: Derived Units

The International System of Units or SI system, by international agreement, has fixed measurement units for seven fundamental properties: length, mass, ...

Collect quantitative data using the International System of Units (SI) or qualitative data as evidence as appropriate to the specialized topic of study;

112.51.c.1.E.

Review and Preview

Data are individual items of information obtained from a population or sample. Data may be classified as qualitative (categorical), quantitative ...

What is a Frequency Distribution

A frequency is the number of times a value of the data occurs. The sum of all the frequency values represents the total number of students included in the ...

Cumulative Frequency Distribution

A cumulative frequency distribution is another type of frequency distribution. Instead of reporting how many data values fall in some classes, it reports ...

Ogive Graph

An ogive graph is sometimes called a cumulative frequency polygon. It is one type of frequency polygon that shows cumulative frequency. In other words, ...

Histogram

The histogram is a graphical representation in the x-y form of data distribution in a data set. The horizontal x-axis is labeled with what the data ...

Relative Frequency Histogram

The relative frequency depicts the proportion of data points that have each value. The frequency tells the number of data points that have each value. ...

Scatter Plot

The most common and easiest way to display the relationship between two variables, x and y, is a scatter plot. A scatter plot shows the direction of a ...

Time-Series Graph

A time-series graph is a line graph with repeated measurements taken at successive intervals of time. It is also called a time series chart. To construct ...

Bar Graph

A bar graph is also called a bar chart and consists of bars that are separated from each other. It either uses horizontal or vertical bars to show ...

Multiple Bar Graph

As the name suggests, a multiple bar graph is the same as a bar graph but has multiple bars to depict relationships between different data values. One can ...

Pareto Chart

A Pareto chart is a bar graph or a combination of both line and bar graphs. The bar lengths represent the individual values or the frequency, while the ...

Pie Chart

A pie chart (or a pie graph) is a circular graphical chart or a pictorial representation of categorical data. It is divided into slices of pie each ...

Organize quantitative or qualitative data using representations appropriate to the specialized topic of study such as scatter plots, line graphs, bar graphs, charts, data tables, diagrams, scientific drawings, and student-prepared models;

112.51.c.1.F.

Modeling and Similitude

A scaled model of a dam is tested in a controlled environment to observe how it will manage water flow, turbulence, and pressure.
This setup predicts the ...

Typical Model Studies

Typical model studies analyze scaled-down systems to predict fluid behavior, focusing on interactions with free surfaces and flow around immersed bodies.
...

design-example-creating-a-hydraulic-model-of-a-dam-spillway

Design Example: Creating a Hydraulic Model of a Dam Spillway

A hydraulic model of a dam spillway is often created by scaling down the design, typically to a 1:15 ratio, to replicate key flow characteristics such as ...

Models, Theories, and Laws

Scientists frequently use models to help them comprehend a specific collection of phenomena. In physics, a model is a condensed version of a physical ...

/files/media/science-education/science-education-thumbs/14686.jpg

Concepts and Prototypes

The human nervous system translates sensory stimuli into neural impulses, which the brain processes into information as thoughts and memories, forming ...

Molecular Models

Physical models representing molecular architectures of chemical compounds play essential roles in understanding chemistry. The use of molecular models ...

Develop and use models to represent phenomena, systems, processes, or solutions to problems as appropriate to the specialized topic of study; and

112.51.c.1.G.

Scientific Laws and Theories

Scientific Laws

In science, a law is defined as a concise, verbal or mathematical, statement that summarizes a vast number of experimental observations. ...

Distinguish among scientific hypotheses, theories, and laws as appropriate to the specialized topic of study.

112.51.c.1.H.

Identify advantages and limitations of models such as their size, scale, properties, and materials as appropriate to the specialized topic of study;

112.51.c.2.A.

Introduction to Statistics

The science of statistics involves collecting, analyzing, interpreting, and presenting data. The method of collecting, organizing, and summarizing data is ...

Statistical Analysis: Overview

An analysis is usually conducted by replicated sampling or repeated measurements on the same sample. This leads to scattered results rather than a single ...

systematic-error-methodological-and-sampling-errors

Systematic Error: Methodological and Sampling Errors

Systematic errors, depending on their source, are of four types &#8211; sampling, instrumental, method, and personal errors.
Sampling errors occur due to ...

Random Error

Indeterminate or random errors arise from several uncontrollable variables in successive measurements.
Since these errors can neither be predicted nor ...

types-of-errors-detection-and-minimization

Types of Errors: Detection and Minimization

Error is the deflection of an obtained result from the expected or true results of an experiment. This happens due to the uncertainty associated with the ...

Realism in Experimentation

Source: Laboratories of Gary Lewandowski, Dave Strohmetz, and Natalie Ciarocco&#8212;Monmouth University

In an ideal world researchers would conduct ...

Bias

Bias refers to any tendency that prevents a question from being considered unprejudiced. In research, bias occurs when one outcome or answer is selected ...

Analyze data appropriate to the specialized topic of study by identifying significant statistical features, patterns, sources of error, and limitations;

112.51.c.2.B.

Scientific and engineering practices. The student, for at least 40% of instructional time, asks questions, identifies problems, and plans and safely conducts classroom, laboratory, and field investigations to explain phenomena or design solutions using appropriate tools and models. The student is expected to:

Accuracy and Approximation in Calculations

Description

Transcript

Tags

More videos by Standards