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

Doğrusal bir yaya kuvvet uygulandığında, geri kazandırıcı kuvvet yer değiştirme ile eşit şekilde artar. Bu ilişki, kuvvet–yer değiştirme eğrisinden yapılan işin hesaplanmasını mümkün kılan Hooke yasasını takip eder.Doğrusal olmayan yay için Hooke yasası geçerli değildir. Geri kazandırıcı kuvveti konumla değişir, böylece kuvvet–yer değiştirme grafiği eğri hale gelir.Yayın gerilmesinde yapılan toplam çalışma, bu kuvvetin yer değiştirme üzerindeki entegrasyonuyla belirlenir.Ancak, integral doğrudan çözülür çünkü bir karekök terimi standart entegrasyon sürecini karmaşıklaştırır.İntralı basitleştirmek için yerine koyma kuralı kullanılır. Karmaşık terim için yeni bir değişken tanıtır. Değişkeni ayırt etmek ve türevi yeniden düzenlemek, integral içindeki diğer terimleri değiştirmemize yardımcı olur.Yerine geçmenin uygulanması, karekök içindeki terimi daha basit bir ifadeye dönüştürür ve integralı tek bir kuvvet fonksiyonuna dönüştürür.Entegrasyondan sonra, yeni değişken orijinal yer değiştirme değişkeniyle değiştirilir ve fiziksel yorumu korunur.Son ifade, doğrusal olmayan bir yayı germek için gereken çalışmayı temsil eder.

belirsiz İntegrasyonlarda uygulanan yerine koyma kuralı

Do&#x11F;rusal bir yaya bir kuvvet uyguland&#x131;&#x11F;&#x131;nda, geri getirme kuvveti yer de&#x11F;i&#x15F;tirme ile orant&#x131;l&#x131; olarak artar. Bu davran&#x131;&#x15F;, Hooke Yasas&#x131; ile tan&#x131;mlan&#x131;r ve bu yasa, yay &#xFC;zerinde yap&#x131;lan i&#x15F;in kuvvet-yer de&#x11F;i&#x15F;tirme ili&#x15F;kinden do&#x11F;rudan belirlenmesine olanak tan&#x131;r. Bu durumda kuvvet basit ve &#xF6;ng&#xF6;r&#xFC;lebilir bir &#x15F;ekilde de&#x11F;i&#x15F;ti&#x11F;inden, hesaplama g&#xF6;rece kolayd&#x131;r.Buna kar&#x15F;&#x131;l&#x131;k, do&#x11F;rusal olmayan bir yay Hooke Yasas&#x131;&#x2019;na uymaz. Bu t&#xFC;r bir yayda geri getirme kuvveti konuma orant&#x131;s&#x131;z bir &#x15F;ekilde ba&#x11F;l&#x131;d&#x131;r; dolay&#x131;s&#x131;yla kuvvet-yer de&#x11F;i&#x15F;tirme ili&#x15F;kisi do&#x11F;rusal de&#x11F;il, e&#x11F;risel olur. Bu t&#xFC;r bir geri getirme kuvvetine &#xF6;rnek olarak &#x15F;u ifade verilebilir:\begin{equation*}F \varpropto x \bm{\sqrt{{1 + \left(\jfrac{x}{L}\right)^2}}}\end{equation*}Bu durumda kuvvet, yer de&#x11F;i&#x15F;tirmenin tek ba&#x15F;&#x131;na artt&#x131;&#x11F;&#x131;na k&#x131;yasla daha h&#x131;zl&#x131; artar. Bu ifade, yay daha fazla gerildik&#xE7;e do&#x11F;rusal olmayan etkilerin nas&#x131;l daha belirgin hale geldi&#x11F;ini ortaya koyar.Do&#x11F;rusal olmayan bir yay&#x131; germek i&#xE7;in yap&#x131;lan toplam i&#x15F;, kuvvetin yer de&#x11F;i&#x15F;tirme boyunca toplanmas&#x131;yla belirlenir. Ancak karek&#xF6;k i&#xE7;eren terim, standart t&#xFC;mlev alma y&#xF6;ntemlerini karma&#x15F;&#x131;kla&#x15F;t&#x131;rd&#x131;&#x11F;&#x131; i&#xE7;in bu niceli&#x11F;in do&#x11F;rudan hesaplanmas&#x131; zordur. Bu nedenle hesaplamay&#x131; basitle&#x15F;tirmek amac&#x131;yla de&#x11F;i&#x15F;ken d&#xF6;n&#xFC;&#x15F;&#xFC;m&#xFC; (yerine koyma) kural&#x131; uygulan&#x131;r. Bu y&#xF6;ntem, karek&#xF6;k i&#xE7;indeki terimi temsil edecek yeni bir de&#x11F;i&#x15F;ken tan&#x131;mlayarak kalan terimlerin daha kolay ele al&#x131;nmas&#x131;na olanak tan&#x131;r.Yerine konulan de&#x11F;i&#x15F;kenin t&#xFC;revi al&#x131;narak ve gerekli cebirsel d&#xFC;zenlemeler yap&#x131;larak ifade, tek bir kuvvet (&#xFC;s) terimine indirgenmi&#x15F; bi&#xE7;imde daha basit bir formda yeniden yaz&#x131;labilir. T&#xFC;mlev alma i&#x15F;lemi tamamland&#x131;ktan sonra, elde edilen sonucun fiziksel anlam&#x131;n&#x131; korumak amac&#x131;yla yerine konulan de&#x11F;i&#x15F;ken tekrar orijinal yer de&#x11F;i&#x15F;tirme de&#x11F;i&#x15F;keni ile ifade edilir. Ortaya &#xE7;&#x131;kan son ifade, do&#x11F;rusal olmayan yay&#x131; germek i&#xE7;in yap&#x131;lan i&#x15F;i temsil eder.

Belirsiz İntegrasyonlarda Uygulanan Yerine Koyma Kuralı

Do&#x11F;rusal bir yaya bir kuvvet uyguland&#x131;&#x11F;&#x131;nda, geri getirme kuvveti yer de&#x11F;i&#x15F;tirme ile orant&#x131;l&#x131; olarak ...

substitution-rule-applied-to-indefinite-integrals

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Integrals

İntegraller

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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