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

선형 스프링에 힘을 가하면 복원력은 변위에 따라 고르게 증가합니다. 이 관계는 힘-변위 곡선에서 수행된 일의 계산을 가능하게 하는 후크의 법칙을 따릅니다.비선형 스프링의 경우, 후크의 법칙은 적용되지 않습니다. 복원력은 위치에 따라 변하므로, 힘-변위 그래프는 곡선이 됩니다.스프링을 늘릴 때 이루어지는 총 일은 이 힘을 변위에 대해 적분하여 결정됩니다.그러나 적분은 제곱근 항이 표준 적분 과정을 복잡하게 만들기 때문에 직접적으로 푸는 것은 어렵습니다.적분을 단순화하기 위해 치환 규칙을 사용합니다. 복소수 항에 새로운 변수를 도입합니다. 변수를 미분하고 미분을 재배열하면 적분 내 다른 항들을 대체할 수 있습니다.치환을 적용하면 제곱근 내의 항이 더 단순한 표현식으로 변환되어 적분이 단일 멱함수로 변환됩니다.적분 후에는 새로운 변수를 원래의 변위 변수로 대체하여 물리적 해석을 유지합니다.최종 식은 비선형 스프링을 늘리는 데 필요한 작업을 나타냅니다.

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&#xC120;&#xD615; &#xC6A9;&#xC218;&#xCCA0;&#xC5D0; &#xD798;&#xC774; &#xAC00;&#xD574;&#xC9C0;&#xBA74; &#xBCF5;&#xC6D0;&#xB825;&#xC740; ...

substitution-rule-applied-to-indefinite-integrals

Education

교육

JoVE Core

Calculus

미적분학

Unit

단위

Integrals

적분

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Substitution Rule Applied to Indefinite Integrals

무한적분에 적용되는 치환 규칙

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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