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

En los cálculos de ingeniería, la precisión, los límites y la aproximación se utilizan para representar un valor.
Considere dos bloques cuadrados de diferentes dimensiones. Los valores precisos de sus áreas se pueden obtener redondeando sus números a los valores más cercanos basados en cifras significativas.
Considere un cono circular recto que tiene un radio de base, r, y una altura, h. Si el cono se corta circularmente a una distancia x de su vértice, de modo que el elemento cortado tiene un grosor, Δx, entonces se puede determinar su volumen.
Aplicando el límite de Δv a dv y de Δx a dx, y despreciando los diferenciales de orden superior, se obtiene la expresión del volumen.
Considere un arco AB, subtendiendo un pequeño ángulo en O. La longitud del arco se puede aproximar a la base de un triángulo rectángulo.
Si la longitud de la hipotenusa es la unidad, la longitud del arco es igual a sine theta, aproximadamente igual a theta. Además, el coseno theta también es aproximadamente igual a la unidad.
Para theta de un grado, el valor aproximado para sen theta y tan theta es casi el mismo.

<ol>
	<li>Meriam, J.L.; Kraige, L.G., y Bolton, J.N. (2020). Ingeniería Mecánica \u2012 Estática. Hoboken, Nueva Jersey: John Wiley. págs. – 13-14.</li>
</ol>

exactitud, límites y aproximaciones

Exactitud, límites y aproximaciones son comunes en muchos campos, especialmente en cálculos de ingeniería. Estos conceptos son imperativos para asegurar que un valor dado esté lo más cerca posible de su valor real. La exactitud se define como la cercanía del valor medido al valor verdadero o real. En la mecánica de ingeniería, se toman medidas repetidas durante análisis teóricos o experimentales para asegurar que el resultado sea preciso y exacto. La exactitud de cualquier solución se basa en los datos empíricos y los resultados obtenidos de diferentes cálculos o mediciones. Consideremos un contenedor cilíndrico lleno de líquido. El volumen del líquido se puede determinar midiendo las dimensiones del contenedor y calculando el volumen usando una fórmula apropiada. Sin embargo, en la práctica, es imposible medir las dimensiones del contenedor con precisión absoluta. Las aproximaciones son particularmente útiles cuando se trata de funciones complejas que son difíciles de resolver analíticamente. El grado de aproximación se refiere al número de cifras significativas utilizadas en las mediciones. Cuantas más cifras significativas se utilicen, más precisa será el resultado. Si bien las aproximaciones son útiles, se debe tener cuidado al usarlas, ya que pueden introducir errores que afecten la exactitud del cálculo. Esto significa que es esencial comprender el error introducido por la aproximación y cómo puede afectar el resultado. Por ejemplo, para un ángulo relativamente pequeño, generalmente menor o igual a 15° o 0,26 radianes, se pueden utilizar aproximaciones para simplificar las funciones de seno, coseno y tangente. Cabe destacar que la exactitud de estas aproximaciones disminuye cuando el ángulo es mayor que 15° o 0,26 radianes. Sin embargo, para ángulos menores o iguales a 15° (0,26 radianes), estas fórmulas tienen buena exactitud y producen valores casi iguales al valor exacto. El concepto de límites es igualmente importante en los cálculos de ingeniería. Los límites se utilizan para determinar el comportamiento de una función, especialmente cuando una variable independiente se acerca a un valor específico. Consideremos una función que se acerca al infinito a medida que x se acerca a un cierto valor. El límite define el valor máximo que se puede alcanzar sin superar el valor especificado de x. Los límites son particularmente útiles para determinar la estabilidad y el rendimiento de sistemas en aplicaciones críticas.

Exactitud, límites y aproximaciones

La precisión, los límites y las aproximaciones son comunes en muchos campos, especialmente en los cálculos de ingeniería. Estos conceptos son imperativos para garantizar que un ...

Education

Educación

JoVE Core

Mechanical Engineering

Ingeniería Mecánica

Unit 1

Unidad 1

An Introduction to Statics

Introducción a la Estática

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

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

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

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Chemistry and Physics

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112.51.c.1.B.

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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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