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

Nos cálculos de engenharia, precisão, limites e aproximação são usados para representar um valor.
Considere dois blocos quadrados de dimensões diferentes. Os valores precisos de suas áreas podem ser obtidos arredondando seus números para os valores mais próximos com base em algarismos significativos.
Considere um cone circular reto com um raio de base, r, e altura, h. Se o cone for cortado circularmente a uma distância x de seu vértice, de modo que o elemento fatiado tenha uma espessura, Δx, então seu volume pode ser determinado.
Aplicando o limite de Δv a dv e Δx a dx, e negligenciando diferenciais de ordem superior, a expressão para o volume é obtida.
Considere um arco AB, subtendendo um pequeno ângulo em O. O comprimento do arco pode ser aproximado a uma base de um triângulo retângulo.
Se o comprimento da hipotenusa for unidade, o comprimento do arco é igual a sine, aproximadamente igual a. Além disso, cosseno também é aproximadamente igual a unidade.
Para de um grau, o valor aproximado para sin theta e tan theta é quase o mesmo.

<ol>
	<li>Oliveira, J.L.; Kraige, L.G., e Bolton, J.N. (2020). Engenharia Mecânica \u2012 Estática. Hoboken, Nova Jersey: John Wiley. pp – 13-14.</li>
</ol>

precisão, limites e aproximações

Precisão, limites e aproximações são comuns em muitas áreas, especialmente em cálculos de engenharia. Esses conceitos são imperativos para garantir que determinado valor esteja o mais próximo possível do seu valor verdadeiro.

Precisão é definida como a proximidade do valor medido em relação ao valor verdadeiro ou real. Na engenharia mecânica, medições repetidas são feitas durante análises teóricas ou experimentais para garantir que o resultado seja preciso e exato.

A precisão de qualquer solução é baseada em dados empíricos e nos resultados obtidos a partir de diferentes cálculos ou medições. Considere um recipiente cilíndrico cheio de líquido. O volume do líquido pode ser determinado medindo as dimensões do recipiente e calculando o volume usando uma fórmula apropriada. No entanto, na prática, é impossível medir as dimensões do recipiente com precisão absoluta.

As aproximações são especialmente úteis ao lidar com funções complexas que são desafiadoras de resolver analiticamente. O grau de aproximação refere-se ao número de algarismos significativos usados nas medições. Quando mais algarismos significativos são usados, o resultado é mais preciso. Embora as aproximações sejam úteis, é preciso ter cuidado ao usá-las, pois elas podem introduzir erros que afetam a precisão do cálculo. Isso significa que é essencial entender o erro introduzido pela aproximação e como ele pode afetar o resultado.

Por exemplo, para um ângulo relativamente pequeno, geralmente menor ou igual a 15º, ou 0,26 radianos, aproximações podem ser usadas para simplificar as funções seno, cosseno e tangente. Notavelmente, a precisão dessas aproximações diminui quando o ângulo é maior que 15 ou 0,26 radianos. No entanto, para ângulos menores ou iguais a 15⁠º (0,26 radianos), essas fórmulas têm boa precisão e produzem valores quase iguais ao valor exato.

O conceito de limites é igualmente importante em cálculos de engenharia. Limites são usados para determinar o comportamento de uma função, especialmente quando uma variável independente se aproxima de um valor específico. Considere uma função aproximada do infinito à medida que x se aproxima de determinado valor. O limite define o valor máximo que pode ser alcançado sem ultrapassar o valor especificado de x. Limites são especialmente úteis para determinar a estabilidade e o desempenho de sistemas em aplicações críticas.

Precisão, limites e aproximações

Precisão, limites e aproximações são comuns em muitos campos, especialmente em cálculos de engenharia. Esses conceitos são imperativos para garantir que um ...

Education

JoVE Core

Mechanical Engineering

Engenharia Mecânica

Unit 1

Unidade 1

An Introduction to Statics

Uma Introdução à Estática

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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

O JoVE Core Mechanical Engineering oferece vídeos animados concisos e de fácil compreensão que abrangem tópicos-chave como equilíbrio, forças, análise de treliças e mecânica estrutural. Explore suas aplicações no mundo real e observe esses conceitos em ação por meio de experimentos conduzidos por cientistas.

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

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Deductive Reasoning

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Correlation and Causation

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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

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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

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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, ...

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Guidelines in Case of a Laboratory Emergency

Robert M. Rioux &amp; Zhifeng Chen, Pennsylvania State University, University Park, PA
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Safe Handling of Mineral Acids

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Handling Chemical Spills

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Proper Use of Autoclaves

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Fume Hoods and Laminar Flow Cabinets

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Fume hoods and laminar flow cabinets are engineering controls ...

Decontamination for Laboratory Biosafety

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

	Preparation of LaboratoryExpand
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	Determining the Density of an EggExpand
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Proper Lab Notebook Keeping

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

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

	Nitration of Methyl BenzoateExpand
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Use tools appropriate to the specialized topic of study;

112.51.c.1.D.

SI Units: 2019 Redefinition

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International System of Units

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Measurement: Standard Units

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Measurement: Derived Units

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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

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What is a Frequency Distribution

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Cumulative Frequency Distribution

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Ogive Graph

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Histogram

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Relative Frequency Histogram

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Scatter Plot

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Time-Series Graph

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Bar Graph

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Multiple Bar Graph

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Pareto Chart

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Pie Chart

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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

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...

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

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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

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

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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.

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Realism in Experimentation

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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

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