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Rescuing 1 sources and tagging 0 as dead.) #IABot (v2.0.1

नया पृष्ठ

यहाँ पर विभिन्न आकार-प्रकार के पिण्डों के [[जड़त्वाघूर्ण]] दिये गये हैं। ज। दत्वाघूर्ण की इकाई की [[विमा]] द्रव्यमान × लम्बाई2 होती है। नीचे दिये गये व्यंजकों की गणना में यह माना गया है कि [[घनत्व]] सर्वत्र समान है।

'''टिप्पणी''': जहाँ कहीं भी अलग से न कहा गया हो, यह माना गया है कि घूर्णन-अक्ष [[संहति-केन्द्र|द्रव्यमान केन्द्र]] से जाता है।

{|class="wikitable"
|-
! वर्णन || चित्र || जड़त्वाघूर्ण || टिप्पणी
|-
| पतला बेलनाकार शेल (shell) जिसके सिरे खुले हैं, त्रिज्या ''r'' तथा द्रव्यमान ''m''
| [[चित्र:moment of inertia thin cylinder.png]] || <math>I = m r^2 \,\!</math> || शेल की मोटाई को नगण्य (शून्य) माना गया है। यह अगले पिण्ड का एक विशेष दशा है जिसमें ''r''1=''r2.
|-
| मोटी दिवार वाली खुले सिरों वाली बेलनाकार नली जिसकी अन्त:त्रिज्या ''r''1, वाह्य त्रिज्या ''r''2, लम्बाई ''h'' एवं द्रव्यमान ''m''
| [[चित्र:moment of inertia thick cylinder h.png]]
| <math>I_z = \frac{1}{2} m\left({r_1}^2 + {r_2}^2\right)</math><ref>[http://www.livephysics.com/problems-and-answers/classical-mechanics/find-moment-of-inertia-of-a-uniform-hollow-cylinder.html Classical Mechanics - Moment of inertia of a uniform hollow cylinder] {{Webarchive|url=https://web.archive.org/web/20080207072800/http://www.livephysics.com/problems-and-answers/classical-mechanics/find-moment-of-inertia-of-a-uniform-hollow-cylinder.html |date=7 फ़रवरी 2008 }}. LivePhysics.com. Retrieved on [[2008-01-31]].</ref> <math>I_x = I_y = \frac{1}{12} m\left[3\left({r_2}^2 + {r_1}^2\right)+h^2\right]</math> or when defining the normalized thickness ''tn'' = ''t''/''r'' and letting ''r'' = ''r''2, then <math>I_z = mr^2\left(1-t_n+\frac{1}{2}t_n^2\right) </math>
| घनत्व ''ρ'' और उपरोक्त ज्यामिति के साथ <math>I_z = \frac{1}{2} \pi\rho h\left({r_2}^4 - {r_1}^4\right)</math>
|-
|''r'' त्रिज्या, ''h'' उंचाई, तथा ''m'' द्रव्यमान का ठोस बेलन
| [[चित्र:moment of inertia solid cylinder.svg|170px]]||<math>I_z = \frac{m r^2}{2}\,\!</math> <math>I_x = I_y = \frac{1}{12} m\left(3r^2+h^2\right)</math>|| This is a special case of the previous object for ''r''1=0.
|-
|Thin, solid [[disk (mathematics)|disk]] of radius ''r'' and mass ''m''||[[चित्र:moment of inertia disc.svg|170px]]||<math>I_z = \frac{m r^2}{2}\,\!</math> <math>I_x = I_y = \frac{m r^2}{4}\,\!</math>|| This is a special case of the previous object for ''h''=0.
|-
|Thin circular [[hoop]] of radius ''r'' and mass ''m''||[[चित्र:moment of inertia hoop.svg|170px]]||<math>I_z = m r^2\!</math> <math>I_x = I_y = \frac{m r^2}{2}\,\!</math>|| This is a special case of a [[torus]] for ''b''=0. (See below.)
|-
|Solid [[sphere]] of radius ''r'' and mass ''m''||[[चित्र:moment of inertia solid sphere.svg|170px]]||<math>I = \frac{2 m r^2}{5}\,\!</math>
|A sphere can be taken to be made up of a stack of infinitesimal thin, solid discs, where the radius differs from 0 to r.
|-
|Hollow [[sphere]] of radius ''r'' and mass ''m''||[[चित्र:moment of inertia hollow sphere.svg|170px]]||<math>I = \frac{2 m r^2}{3}\,\!</math>||Similar to the solid sphere, only this time considering a stack of infinitesimal thin, circular hoops.
|-
|[[Oblate spheroid|Oblate Spheroid]] of major ''a'', minor ''b'' and mass ''m''||||<math>I = \frac{2 m b^2}{3}\,\!</math>||—
|-
|[[right angle|Right]] circular [[cone (geometry)|cone]] with radius ''r'', height ''h'' and mass ''m''||[[चित्र:moment of inertia cone.svg|120px]]||<math>I_z = \frac{3}{10}mr^2 \,\!</math> <math>I_x = I_y = \frac{3}{5}m\left(\frac{r^2}{4}+h^2\right) \,\!</math>||—
|-
|Solid [[cuboid]] of height ''h'', width ''w'', and depth ''d'', and mass ''m''||[[चित्र:moment of inertia solid rectangular prism.png]]||<math>I_h = \frac{1}{12} m\left(w^2+d^2\right)</math> <math>I_w = \frac{1}{12} m\left(h^2+d^2\right)</math> <math>I_d = \frac{1}{12} m\left(h^2+w^2\right)</math>||For a similarly oriented [[cube (geometry)|cube]] with sides of length <math>s</math>, <math>I_{CM} = \frac{m s^2}{6}\,\!</math>.
|-
|Thin rectangular plane of height ''h'' and of width ''w'' and mass ''m''||[[चित्र:Recplane.svg|250px]]||<math>
I_c = \frac {m(h^2 + w^2)}{12}\,\!</math>||—
|-
|Thin rectangular plane of height ''h'' and of width ''w'' and mass ''m'' (Axis of rotation at the end of the plate)
||[[चित्र:Recplaneoff.svg|250px]]||<math>I_e = \frac {m h^2}{3}+\frac {m w^2}{12}\,\!</math>||—
|-
|[[Rod]] of length ''L'' and mass ''m''||[[चित्र:moment of inertia rod center.png]]||<math>I_{\mathrm{center}} = \frac{m L^2}{12} \,\!</math>||This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a special case of the previous object for ''w''=''L'' and ''h''=''d''=''0''.
|-
|[[Rod]] of length ''L'' and mass ''m'' (Axis of rotation at the end of the rod)||[[चित्र:moment of inertia rod end.png]]||<math>I_{\mathrm{end}} = \frac{m L^2}{3} \,\!</math>||This expression assumes that the rod is an infinitely thin (but rigid) wire.
|-
|[[Torus]] of tube radius ''a'', cross-sectional radius ''b'' and mass ''m''.||[[चित्र:torus cycles.png|122px]]||About a diameter: <math>\frac{1}{8}\left(4a^2 + 5b^2\right)m</math> About the vertical axis: <math>\left(a^2 + \frac{3}{4}b^2\right)m</math>||—
|-
|Plane [[polygon]] with vertices <math>\vec{P}_{1}</math>, <math>\vec{P}_{2}</math>, <math>\vec{P}_{3}</math>, ..., <math>\vec{P}_{N}</math> and mass <math>m</math> uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.||[[चित्र:Polygon moment of inertia.png|130px]]||<math>I=\frac{m}{6}\frac{\sum\limits_{n=1}^{N}\|\vec{P}_{n+1}\times\vec{P}_{n}\|(\vec{P}^{2}_{n+1}+\vec{P}_{n+1}\cdot\vec{P}_{n}+\vec{P}_{n}^{2})}{\sum\limits_{n=1}^{N}\|\vec{P}_{n+1}\times\vec{P}_{n}\|}</math>||—
|}

== इन्हें भी देखें ==
* [[समान्तर अक्ष का प्रमेय]] (Parallel axis theorem)
* [[क्षेत्राघूर्णों की सूची]] (List of area moments of inertia)
* [[जड़त्वाघूर्ण टेंसरों की सूची]] (List of moment of inertia tensors)

== सन्दर्भ ==
<references/>

[[श्रेणी:यान्त्रिकी]]

जड़त्वाघूर्णों की सूची - अवतरण इतिहास

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