Name: work done on a nonlinear spring
Uploaded: 2026-01-12T12:24:06.000Z
Duration: 1 min 27 s
Channel: JoVE

عندما تطبق قوة على نابض خطي، تزداد قوة الاستعادة بشكل متساو مع الإزاحة. تتبع هذه العلاقة قانون هوك، الذي يمكن حساب العمل المنجز من منحنى القوة-الإزاحة.بالنسبة للنابض غير الخطي، لا ينطبق قانون هوك. تتغير قوة الاستعادة مع الموقع، لذا يصبح رسم القوة-الإزاحة منحنيا.يتم تحديد العمل الكلي المبذول في شد النابض من خلال دمج هذه القوة على الإزاحة.ومع ذلك، يصعب حل التكامل مباشرة لأن مصطلح الجذر التربيعي يعقد عملية التكامل القياسية.لتبسيط التكامل، تستخدم قاعدة الاستبدال. يقدم متغيرا جديدا للمصطلح المركب. تفاضل المتغير وإعادة ترتيب المشتقة يساعدنا على استبدال الحدود الأخرى داخل التكامل.تطبيق الاستبدال يحول المصطلح داخل الجذر التربيعي إلى تعبير أبسط، حيث يحول التكامل إلى دالة قوة واحدة.بعد التكامل، يتم استبدال المتغير الجديد بمتغير الإزاحة الأصلي، مع الحفاظ على تفسيره الفيزيائي.التعبير النهائي يمثل العمل المطلوب لتمديد نابض غير خطي.

قاعدة التعويض المطبقة على التكاملات غير المحددة

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substitution-rule-applied-to-indefinite-integrals

Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Integrals

التكاملات

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

Substitution Rule Applied to Indefinite Integrals

work done on a nonlinear spring

Work done on a nonlinear spring can be found by using integration and the substitution rule. A linear spring follows Hooke’s law, so its restoring force increases in direct proportion to displacement. That makes the force-displacement graph a straight line and the work calculation straightforward.

A nonlinear spring does not obey Hooke’s law. Its restoring force changes with position in a non-proportional way, so the force-displacement graph is curved instead of linear. In the example shown, the force increases more rapidly than displacement alone, especially as the spring is stretched farther.

To find the total work, the force must be accumulated over the displacement. Direct integration is difficult because the square root term makes the expression harder to solve with standard methods. The substitution rule simplifies the integral by replacing the expression inside the square root with a new variable.

After the new variable is chosen, it is differentiated and rearranged so the integral becomes a simpler power expression. The calculation is then completed with integration. Finally, the substituted variable is changed back to the original displacement variable, and the result gives the work required to stretch the nonlinear spring.

Work Done on a Nonlinear Spring

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Work done on a nonlinear spring can be found by using integration and the substitution rule. A linear spring follows Hooke’s law, so its restoring force increases in direct proportion to displacement. That makes the force-displacement graph a straight line and the work calculation straightforward. A nonlinear spring does not obey Hooke’s law. Its restoring force changes with position in a non-proportional way, so the force-displacement graph is curved instead of linear. In the example shown,...

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key terms and concepts

Learn how integration and the substitution rule are used to calculate the work required to stretch a nonlinear spring when the force is not linear.

Work Done on a Nonlinear Spring | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Work Done on a Nonlinear Spring

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