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

في الحسابات الهندسية ، يتم استخدام الدقة والحدود والتقريب لتمثيل قيمة.
ضع في اعتبارك كتلتين مربعتين بأبعاد مختلفة. يمكن الحصول على القيم الدقيقة لمناطقهم عن طريق تقريب أرقامهم إلى أقرب القيم بناء على أرقام معنوية.
ضع في اعتبارك مخروطا دائريا يينا نصف قطر قاعدة ، ص ، وارتفاعه ، h. إذا تم تقطيع المخروط بشكل دائري على مسافة x من رأسه ، بحيث يكون للعنصر المقطع سمك ، Δx ، فيمكن تحديد حجمه.
من خلال تطبيق الحد من Δv إلى dv و Δx إلى dx ، وإهمال الفروق ذات الترتيب الأعلى ، يتم الحصول على تعبير الحجم.
ضع في اعتبارك قوس AB ، مع إدراج زاوية صغيرة عند O. يمكن تقريب طول القوس إلى قاعدة مثلث قائم الزاوية.
إذا كان طول الوتر هو الوحدة ، فإن طول القوس يساوي جيب ثيتا ، وهو ما يساوي تقريبا ثيتا. بالإضافة إلى ذلك ، فإن جيب التمام ثيتا يساوي الوحدة تقريبا.
بالنسبة لثيتا بدرجة واحدة ، فإن القيمة التقريبية لثيتا الخطيئة وتان ثيتا هي نفسها تقريبا.

<ol>
	<li>مريم ، جي إل. كريج ، إل جي ، وبولتون ، ج. (2020). الميكانيكا الهندسية - الإحصاء. هوبوكين ، نيو جيرسي: جون وايلي. ص - 13-14.</li>
</ol>

الدقة والنهايات والتقريبات.

<div style="direction:rtl">تعد الدقة والنهايات والتقريبات من المفاهيم الشائعة في العديد من المجالات، خاصة في الحسابات الهندسية. هذه المفاهيم ضرورية لضمان أن تكون القيمة المعطاة قريبة قدر الإمكان من قيمتها الحقيقية. تُعرّف الدقة بأنها قرب القيمة المقاسة من القيمة الحقيقية أو الفعلية. في الميكانيكا الهندسية، يتم أخذ قياسات متكررة خلال التحليلات النظرية أو التجريبية لضمان أن النتيجة دقيقة ومضبوطة. تعتمد دقة أي حل على البيانات التجريبية والنتائج المستخلصة من مختلف الحسابات أو القياسات. لنأخذ على سبيل المثال وعاء أسطواني مملوء بسائل. يمكن تحديد حجم السائل من خلال قياس أبعاد الوعاء وحسابه باستخدام المعادلة المناسبة. ومع ذلك، من الناحية العملية، لا يمكن قياس أبعاد الوعاء بدقة مطلقة. تعد التقريبات مفيدة بشكل خاص عند التعامل مع الدوال المعقدة التي يصعب حلها بشكل تحليلي. تشير درجة التقريب إلى عدد الأرقام المهمة المستخدمة في القياس. كلما زاد عدد الأرقام المهمة المستخدمة، زادت دقة النتيجة. على الرغم من اهمية التقريبات، يجب توخي الحذر عند استخدامها، حيث يمكن أن تؤدي إلى أخطاء تؤثر على دقة الحساب. وهذا يعني أنه من الضروري فهم الخطأ المتوقع من عملية التقريب وكيفية تأثيره على النتيجة النهائية. على سبيل المثال، بالنسبة لزاوية صغيرة نسبيًا، عادةً أقل من أو تساوي ١٥° أو ٠.٢٦ راديان، يمكن استخدام التقريبات لتبسيط دوال الجيب وجيب التمام والظل. من الجدير بالذكر أن دقة هذه التقريبات تقل عندما تكون الزاوية أكبر من ١٥° أو ٠.٢٦ راديان. ومع ذلك، عندما تكون الزاوية أقل من أو تساوي ١٥° (٠.٢٦ راديان)، فإن هذه التقريبات تتمتع بدقة جيدة وتنتج قيمًا تكاد تكون مساوية للقيمة الفعلية. يُعد مفهوم النهايات مهمًا بنفس القدر في الحسابات الهندسية. تُستخدم النهايات لتحديد سلوك الدالة، خاصة عندما تقترب المتغيرات المستقلة من قيمة معينة. لنأخذ على سبيل المثال دالة تقترب من اللانهاية عندما يقترب x من قيمة معينة. تحدد النهاية الحد الأقصى الذي يمكن الوصول إليه دون تجاوز القيمة المحددة لـ x. وتعد النهايات مفيدة بشكل خاص في تحديد استقرار وأداء الأنظمة في التطبيقات الحرجة.</div>

الدقة والحدود والتقريب شائعة في العديد من المجالات ، خاصة في الحسابات الهندسية. هذه المفاهيم ضرورية لضمان أن ...

Education

JoVE Core

Mechanical Engineering

الهندسة الميكانيكية

Unit 1

الوحدة الأولى

An Introduction to Statics

مقدمة في علم السكون (الاستاتيكا)

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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

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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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units of measurement

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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

The Scientific Method

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Inductive Reasoning

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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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Problem Definition

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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

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112.51.c.1.C.

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112.51.c.1.D.

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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.

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Relative Frequency Histogram

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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.

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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

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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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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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