СРАВНЕНИЕ РАЗНИЦ ПЕРИОДОВ МАТЕМАТИЧЕСКОГО МАЯТНИКА ПРИ МАЛЫХ И БОЛЬШИХ УГЛАХ

Аннотация

For pendulums tilted at small angles, although small-angle simplifications are useful, for large-angle pendulums this approach does not work and can lead to major problems in computational models of simple pendulums. This article explores the differences between the periods of oscillation of small-angle and large-angle pendulums and shows how to derive the period in both models from the fundamental laws of nature. Also, this paper presents a general method for deriving the elliptic integral from physical phenomena. Finally, the article makes a number of suggestions for expanding the study of mathematical pendulums. Although this article is not intended as a rigorous mathematical proof, it is intended to shed light on the derivation of the exact periods of pendulums and to explain in detail the mathematics involved in this process. Keywords: pendulum, period, angle, mathematical pendulum. Language: Uzbek Citation: Khudoyberdiyeva , M., Boronov , M., & Mukhammadjonov , A. A. (2025). COMPARISON OF THE DIFFERENCES BETWEEN THE PERIODS OF A MATHEMATICAL PENDULUM AT SMALL AND LARGE ANGLES. Universal International Scientific Journal, 2(1), 11– 22. https://doi.org/10.69891/3060-4540.2025.78.33.001 Doi: https://doi.org/10.5281/zenodo.14636182 Google scholar: https://inlibrary.uz/index.php/universaljurnal/article/view/60750 Crosreff doi: https://doi.org/10.69891/3060-4540.2025.78.33.001 I. Kirish yuborilganda tebranadi. Mayatniklarni Matematik mayatniklar faqat bir uchi birinchi o‘rganganlardan biri bo‘lgan Galileo mustahkamlangan ip va boshqa uchiga Galiley ularning harakati og`ish burchakka biriktirilgan yukdan iborat. Ular qo‘yib bog‘liq emasligini aniqlagan. U mayatniklar 12 Universaljurnal.uz Universal International Scientific Journal 2025, 2(1) tebranishini kuzatish jarayonida mayatniklar — biri taxminiy davrni, ikkinchisi esa aniq maksimal burchak og‘ishiga nisbatan davrni ifodalaydi — va ularni solishtirish izoxronik ekanligini ko‘rgan: ya’ni, orqali taqribiy hisoblash bilan bog‘liq mayatnikning tebranishi davomida vertikalga xatoliklarni aniqlash mumkin. Bu juda muhim, nisbatan qanday burchak hosil qilishidan chunki matematik mayatnikning harakati qat’iy nazar, uning davri har doim bir xil davriy bo‘lganligi sababli, modellashtirishda bo‘ladi. davr vaqtidagi har qanday kichik farq bir Agar bu haqiqat bo‘lsa, unda mayatniklar necha tebranishdan keyin katta xatoliklarga garmonik harakatni namoyon qiladi, bu olib kelishi mumkin. Bu differensial harakatda mexanik sistemani muvozanatga tenglamalarni yechgandan so‘ng, kichik qaytaruvchi kuch, muvozanat vaziyatidan burchak taxminidan nima uchun og‘ish burchagiga mos bo‘ladi. Galileo foydalanilishini ham o‘rganishimiz mumkin. kuzatuvidan beri mayatniklar vaqtni o‘lchash Bu muammoni hal qilish qiyinligiga haqiqatan va umuman ilm-fanda yirik yutuqlarga olib ham ta’sir qiladimi? Agar davrning aniq kelgan, boshqa tarafdan esa Yerning bir kunda yechimi ishlatishga qulay bo‘lsa, unda taqribiy bir marta aylanishini isbotlagan. hisoblash o‘rniga shu aniq yechimdan O‘rta maktab fizikasi kurslarida foydalanish kerak. Agar taqribiy hisoblash mayatniklarni izoxronik deb hisoblashni juda katta xatolik keltirib chiqarmasa va o‘rganamiz. Ularning harakatini ishlatish ancha oson bo‘lsa, undan foydalanish modellashtirishda Galileo taxminiga amal mantiqiydir. Eng asosiysi, ushbu tadqiqot qilamiz va mayatnikning davrini hisoblashda fizika bo‘yicha biz erta bosqichlarda og`ish burchagini hisobga olmaymiz. o‘rgangan umumiy bilimni o‘rganish va uning Darhaqiqat, maktabning dastlabki yillarida bu qanchalik noto‘g‘ri ekanligini aniqlashga natijalarni o‘zimiz ham topa olamiz. Masalan, qaratilgan. men beshinchi sinfda ipga bog‘langan yuvish mashinasi yuvgichlarining tebranish vaqtlarini o‘lchab, oxir-oqibat Galileo bilan bir xil xulosaga kelganimni eslayman. Hayratlanarlisi shundaki, aslida Galileo xato qilgan: mayatnikning maksimal og‘ish burchagi uning tebranish davriga ta’sir qiladi. Nisbatan kichik maksimal og‘ishlarga ega mayatniklar amalda izoxronik bo‘lsa-da, bu taxmin bilan bog‘liq xatolik burchak kattalashgani sari ortib boradi. Ushbu tadqiqotning maqsadi Galileo taxmini qaysi holatlarda to‘g‘ri va noto‘g‘ri 1-rasm. Matematik mayatnik ekanligini tahlil qilishdir. Mayatniklarning ko`rinishi. harakatini asosiy qonunlaridan ikkinchi tartibli differensial tenglama hosil qilish orqali 13 Universaljurnal.uz Universal International Scientific Journal 2025, 2(1) 2-rasm. Erkin matematik mayatnikning davriy tebranish harakati. 𝝎−burchak 𝒅𝜽 tezlik, shuning uchun 𝝎 = . 𝒅𝒕 MATEMATIK Nyuton qonunlarining qo‘llanilishi MAYATNIKLARNI Mayatnikning harakati Nyuton MODELLASHTIRISH qonunlarining aylanma harakatga tatbiq etilishiga asoslanadi: Matematik mayatnik 𝑙 uzunligi 𝑀 = 𝐼𝛼 (1) bo‘lgan va vaznga ega bo`lmagan ipga bu yerda 𝑀 — kuch momenti, I — osilgan, massasi 𝑚 bo‘lgan va moddiy inersiya momenti, α — burchak tezlanish. nuqta deb qarash mumkin bo`lgan yukdan Kuch momenti aylantiruvchi kuchning ibo