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

在工程计算中，精度、限值和近似值用于表示值。
考虑两个不同尺寸的正方形块。可以通过根据有效数字将其数字四舍五入到最接近的值来获得其面积的准确值。
考虑一个具有底半径 r 和高度 h 的直圆锥。如果圆锥体在距其顶点 x 的距离处进行圆形切片，使得切片单元的厚度为 Δx，则可以确定其体积。
通过应用从 Δv 到 dv 和 Δx 到 dx 的极限，并忽略高阶差分，可以得到体积的表达式。
考虑一个圆弧 AB，在 O 处呈现一个小角度。弧长可以近似于直角三角形的底面。
如果斜边的长度是 unity，则弧长等于 sine theta，大约等于 theta。此外，余弦 theta 也近似等于 1。
对于 1 度的 theta，sin theta 和 tan theta 的近似值几乎相同。

<ol>
	<li>梅里亚姆，JL;Kraige， LG 和 Bolton， JN （2020）。工程力学 \u2012 静力学。新泽西州霍博肯:约翰·威利。第 – 13-14 页。</li>
</ol>

准确性、限制和近似

准确性、限制和近似在许多领域都很常见，特别是在工程计算中。这些概念对于确保计算值尽可能接近真实值至关重要。 准确性被定义为测量值与真实值或实际值之间的接近程度。在工程力学中，通过理论或实验分析对计算值重复测量，以确保结果既精确又准确。 任何解的准确性都基于经验数据和从不同计算或测量所得到的结果。例如，考虑一个装有液体的圆柱形容器。可以通过测量容器的尺寸并使用适当的容积计算公式来确定液体的体积。但在实践中，不可能以绝对的精度来测量容器。 近似在处理难以解析求解的复杂函数时非常有用。近似程度通常指所使用的有效数字数量，使用更多有效数字能提高结果的准确性。虽然近似很有用，但在使用时必须注意，近似也可能引入误差，因此必须了解其误差及对结果的影响。 例如，对于相对较小的角度（通常小于或等于15°或0.26弧度），可以使用近似简化正弦、余弦和正切函数。但是当角度大于15°或0.26弧度时，近似的准确性便会降低。然而，对于小于或等于15°（0.26弧度）的角度，近似公式具有较高的准确性，能够得出接近于精确值的值。 在工程计算中，限制的概念同样重要。限制通常用来确定函数的行为，特别是在自变量接近特定的值时。若一个函数在x值接近于某个值时结果将趋向于无限大。限制定义了可以在不超过x指定值的情况下可达到的最大值。限制在确定关键系统的稳定性和性能方面起着至关重要的作用。

精度、极限和近似值在许多领域中都很常见，尤其是在工程计算中。这些概念对于确保 ...

Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

An Introduction to Statics

静力学导论

KEY TERMS AND CONCEPTS

关键术语与概念

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...

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newton's law of motion

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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...

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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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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.

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Physics

PHYS.2

PHYS.2.A

Chemistry and Physics

IPC.2

IPC.2.A

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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.

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ExpandObtain branches containing a few dozen typical leaves from five different local tree species. ...

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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.

Proper Personal Protective Equipment

Robert M. Rioux &#38; William A Elliott, Pennsylvania State University, University Park, PA
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Emergency Eyewash and Shower Stations

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Electrical Safety

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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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