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

W obliczeniach inżynierskich dokładność, granice i przybliżenie są używane do reprezentowania wartości.
Rozważ dwa kwadratowe bloki o różnych wymiarach. Dokładne wartości ich powierzchni można uzyskać, zaokrąglając ich liczby do najbliższych wartości na podstawie liczb znaczących.
Rozważmy prawy okrągły stożek o promieniu podstawy r i wysokości h. Jeśli stożek jest przecięty kołowo w odległości x od jego wierzchołka, tak że przecięty element ma grubość Δx, to można określić jego objętość.
Stosując granicę od Δv do dv i Δx do dx, pomijając różniczki wyższego rzędu, otrzymuje się wyrażenie dla objętości.
Rozważmy łuk AB, podchodzący do małego kąta w O. Długość łuku można przybliżyć do podstawy trójkąta prostokątnego.
Jeśli długość przeciwprostokątnej jest równa jedności, długość łuku jest równa sinusoidalnej theta, w przybliżeniu równej theta. Ponadto cosinus theta również w przybliżeniu równa się jedności.
Dla theta jednego stopnia, przybliżona wartość dla sin theta i tan theta jest prawie taka sama.

<ol>
	<li>Meriam, J.L.; Kraige, LG i Bolton, J.N. (2020). Mechanika inżynierska \u2012 Statyka. Hoboken, New Jersey: John Wiley. str. – 13-14.</li>
</ol>

dokładność, limity i przybliżenia

Dokładność, limity i przybliżenia są powszechne w wielu dziedzinach, zwłaszcza w obliczeniach inżynierskich. Są one niezbędne do zapewnienia, że dana wartość jest jak najbliższa jej rzeczywistej wartości.

Dokładność jest definiowana jako bliskość wartości zmierzonej do wartości prawdziwej lub rzeczywistej. W dziedzinie mechaniki inżynierskiej, podczas teoretycznych lub eksperymentalnych analiz, wykonuje się powtarzane pomiary, aby upewnić się, że wynik jest precyzyjny i dokładny.

Dokładność rozwiązania opiera się na danych empirycznych i wynikach uzyskanych z różnych obliczeń lub pomiarów. Przyjrzyjmy się cylindrycznemu pojemnikowi wypełnionemu cieczą. Objętość cieczy można wyznaczyć, mierząc wymiary pojemnika i obliczając objętość przy użyciu odpowiedniego wzoru. Jednak w praktyce niemożliwe jest dokładne zmierzenie wymiarów tego pojemnika.

Przybliżenia są szczególnie przydatne przy rozwiązywaniu skomplikowanych funkcji, które są trudne do rozwiązania analitycznie. Stopień przybliżenia odnosi się do liczby istotnych cyfr używanych w pomiarach. Im więcej cyfr istotnych jest używanych, tym wynik jest bardziej dokładny. Mimo że przybliżenia są użyteczne, trzeba uważać, korzystając z nich, ponieważ mogą wprowadzać błędy, które wpływają na dokładność obliczeń. Oznacza to, że istotne jest zrozumienie błędu wprowadzonego przez przybliżenie i jak może to wpłynąć na wynik.



Na przykład, dla stosunkowo małego kąta, zazwyczaj mniejszego lub równego 15° lub 0,26 radiana, można używać przybliżeń do uproszczenia funkcji sinus, cosinus i tangens. Warto zauważyć, że dokładność tych przybliżeń maleje, gdy kąt jest większy niż 15° lub 0,26 radiana. Jednak dla kątów mniejszych lub równych 15° (0,26 radiana), te wzory mają dobrą dokładność i dają wartości praktycznie równe wartości rzeczywistej.

Pojęcie granic jest równie ważne w obliczeniach inżynierskich. Granice są używane do określania zachowania się funkcji, zwłaszcza gdy zmienna niezależna zbliża się do określonej wartości. Przyjrzyjmy się funkcji, która dąży do nieskończoności, gdy zmienna x dąży do określonej wartości. Granica definiuje maksymalną wartość, która może zostać osiągnięta bez przekroczenia określonej wartości x. Granice są szczególnie przydatne przy określaniu stabilności i wydajności systemów w krytycznych zastosowaniach.

Dokładność, limity i przybliżenia

Dokładność, granice i przybliżenia są powszechne w wielu dziedzinach, zwłaszcza w obliczeniach inżynierskich. Koncepcje te są niezbędne do zapewnienia, że ...

Education

JoVE Core

Mechanical Engineering

Inżynieria mechaniczna

Unit 1

Jednostka 1

An Introduction to Statics

Wstęp do statyki

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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.

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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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112.51.c.1.C.

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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.

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112.51.c.1.H.

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112.51.c.2.A.

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Accuracy and Approximation in Calculations

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