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

Nei calcoli ingegneristici, l'accuratezza, i limiti e l'approssimazione vengono utilizzati per rappresentare un valore.
Considera due blocchi quadrati di dimensioni diverse. I valori accurati delle loro aree possono essere ottenuti arrotondando i loro numeri ai valori più vicini basati su cifre significative.
Si consideri un cono circolare retto con un raggio di base, r, e un'altezza, h. Se il cono viene tagliato circolarmente a una distanza x dal suo vertice, in modo tale che l'elemento affettato abbia uno spessore Δx, allora il suo volume può essere determinato.
Applicando il limite da Δv a dv e da Δx a dx, e trascurando i differenziali di ordine superiore, si ottiene l'espressione per il volume.
Si consideri un arco AB, che sottende un piccolo angolo in O. La lunghezza dell'arco può essere approssimata alla base di un triangolo rettangolo.
Se la lunghezza dell'ipotenusa è l'unità, la lunghezza dell'arco è uguale al seno theta, approssimativamente uguale al theta. Inoltre, anche il coseno theta è approssimativamente uguale all'unità.
Per theta di un grado, il valore approssimativo per sin theta e tan theta è quasi lo stesso.

<ol>
	<li>Meriam, J.L.; Kraige, LG e Bolton, JN (2020). Ingegneria Meccanica \u2012 Statica. Hoboken, New Jersey: John Wiley. pp – 13-14.</li>
</ol>

precisione, limiti e approssimazioni

La precisione, i limiti e le approssimazioni sono comuni in molti campi, specialmente nei calcoli ingegneristici. Questi concetti sono fondamentali per garantire che un dato valore sia il più vicino possibile al suo valore reale.

La precisione viene definita come la vicinanza del valore misurato al valore vero o effettivo. Nell'ingegneria meccanica, durante le analisi teoriche o sperimentali vengono effettuate misurazioni ripetute per garantire che il risultato sia preciso e accurato.

La precisione di una soluzione dipende dai dati empirici e dai risultati ottenuti con diversi calcoli o misurazioni. Prendiamo ad esempio un contenitore cilindrico riempito con un liquido. Il volume del liquido può essere determinato misurando le dimensioni del contenitore e calcolando il volume attraverso una formula appropriata. Tuttavia, nella pratica, è impossibile misurare le dimensioni del contenitore con assoluta precisione.

Le approssimazioni sono particolarmente utili quando si ha a che fare con funzioni complesse difficili da risolvere analiticamente. Il grado di approssimazione si riferisce al numero di cifre significative utilizzate nelle misurazioni. Quando si utilizzano più cifre significative, il risultato è più accurato. Anche se le approssimazioni sono utili, bisogna fare attenzione nell'usarle, in quanto possono introdurre errori che influiscono sulla precisione del calcolo. Ciò significa che è fondamentale capire l'errore introdotto dall'approssimazione e come può influire sul risultato.

Ad esempio, per un angolo relativamente piccolo, di solito inferiore o uguale a 15° o 0,26 radianti, si possono utilizzare le approssimazioni per semplificare le funzioni seno, coseno e tangente. Si noti che l'accuratezza di queste approssimazioni diminuisce quando l'angolo è maggiore di 15° o 0,26 radianti. Tuttavia, per angoli minori o uguali a 15° (0,26 radianti), queste formule hanno una buona precisione e portano a valori quasi uguali al valore esatto.

Il concetto di limiti è altrettanto importante nei calcoli ingegneristici. I limiti vengono utilizzati per determinare il comportamento di una funzione, in particolare quando una variabile indipendente si avvicina a un valore specifico. Consideriamo una funzione che tende all'infinito quando x si avvicina a un certo valore. Il limite definisce il valore massimo che può essere raggiunto senza superare il valore specificato di x. I limiti sono particolarmente utili per determinare la stabilità e le prestazioni dei sistemi in applicazioni critiche.

Precisione, limiti e approssimazioni

Precisione, limiti e approssimazioni sono comuni in molti campi, specialmente nei calcoli ingegneristici. Questi concetti sono indispensabili per garantire che un ...

Education

Didattica

JoVE Core

Mechanical Engineering

Ingegneria Meccanica

Unit 1

Unità 1

An Introduction to Statics

Introduzione alla statica

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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

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newton's law of motion

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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...

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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...

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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...

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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

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Physics

PHYS.2

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Chemistry and Physics

IPC.2

IPC.2.A

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112.51.c.1.A.

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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.

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Working with Hot and Cold Sources

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112.51.c.1.C.

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Proper Lab Notebook Keeping

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112.51.c.1.D.

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112.51.c.1.E.

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112.51.c.1.F.

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112.51.c.1.G.

Scientific Laws and Theories

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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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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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