实心球体转动惯量公式推导中的疑问 证明：球的转动惯量J=2mR^2/5
\u5b9e\u5fc3\u7403\u4f53\u8f6c\u52a8\u60ef\u91cf\u516c\u5f0f\u63a8\u5bfc\u4e2d\u7684\u7591\u95ee I = \u222b 2/3 r^2 dm = \u222b (0,R) 2/3 r^2 *\u03c1*4\u03c0*\u8fd9\u662f\u5229\u7528\u7403\u58f3\u7684\u8f6c\u52a8\u60ef\u91cf\u79ef\u5206\u7b97\u7684\u3002\u7403\u58f3\u5bf9\u76f4\u5f84\u7684\u8f6c\u52a8\u60ef\u91cf\u662f2/3*R^2*m\u3002
\u76f4\u63a5\u7b97\u7684\u8bdd\uff0c \u8981\u5728\u6c42\u5750\u6807\u91cc\u6c42\u3002I=\u222br^2dm=\u222b(0,2pi)\u222b(0,pi)\u222b(0,R)r^2*\u03c1*r^2*sin(phi)*dr*d(phi)*d(theta)\u3002
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\u8f6c\u52a8\u60ef\u91cf(Moment of Inertia)\u662f\u521a\u4f53\u7ed5\u8f74\u8f6c\u52a8\u65f6\u60ef\u6027\uff08\u56de\u8f6c\u7269\u4f53\u4fdd\u6301\u5176\u5300\u901f\u5706\u5468\u8fd0\u52a8\u6216\u9759\u6b62\u7684\u7279\u6027\uff09\u7684\u91cf\u5ea6\uff0c\u7528\u5b57\u6bcdI\u6216J\u8868\u793a\u3002
\u5728\u7ecf\u5178\u529b\u5b66\u4e2d\uff0c\u8f6c\u52a8\u60ef\u91cf\uff08\u53c8\u79f0\u8d28\u91cf\u60ef\u6027\u77e9\uff0c\u7b80\u79f0\u60ef\u8ddd\uff09\u901a\u5e38\u4ee5I \u6216J\u8868\u793a\uff0cSI \u5355\u4f4d\u4e3a kg\u00b7m²\u3002\u5bf9\u4e8e\u4e00\u4e2a\u8d28\u70b9\uff0cI = mr²\uff0c\u5176\u4e2d m \u662f\u5176\u8d28\u91cf\uff0cr \u662f\u8d28\u70b9\u548c\u8f6c\u8f74\u7684\u5782\u76f4\u8ddd\u79bb\u3002\u8f6c\u52a8\u60ef\u91cf\u5728\u65cb\u8f6c\u52a8\u529b\u5b66\u4e2d\u7684\u89d2\u8272\u76f8\u5f53\u4e8e\u7ebf\u6027\u52a8\u529b\u5b66\u4e2d\u7684\u8d28\u91cf\uff0c\u53ef\u5f62\u5f0f\u5730\u7406\u89e3\u4e3a\u4e00\u4e2a\u7269\u4f53\u5bf9\u4e8e\u65cb\u8f6c\u8fd0\u52a8\u7684\u60ef\u6027\uff0c\u7528\u4e8e\u5efa\u7acb\u89d2\u52a8\u91cf\u3001\u89d2\u901f\u5ea6\u3001\u529b\u77e9\u548c\u89d2\u52a0\u901f\u5ea6\u7b49\u6570\u4e2a\u91cf\u4e4b\u95f4\u7684\u5173\u7cfb\u3002
\u6269\u5c55\u8d44\u6599
\u5706\u76d8\u548c\u5b9e\u5fc3\u7403\u7684\u8f6c\u52a8\u60ef\u91cf\u8ba1\u7b97\u516c\u5f0f\uff1a\u5bf9\u4e8e\u8584\u5706\u76d8\uff0c\u5f53\u56de\u8f6c\u8f74\u901a\u8fc7\u4e2d\u5fc3\u4e0e\u76d8\u9762\u5782\u76f4\u65f6\uff0c I=1/2mr²\u3002
\u5f53\u56de\u8f6c\u8f74\u901a\u8fc7\u8fb9\u7f18\u4e0e\u76d8\u9762\u5782\u76f4\u65f6\uff0c I=3/2mr²\u3002
\u5bf9\u4e8e\u5b9e\u5fc3\u7403\u4f53\uff0c\u5f53\u56de\u8f6c\u8f74\u4e3a\u7403\u4f53\u7684\u4e2d\u5fc3\u8f74\u65f6\uff0cI=2/5mr²\u3002\u5f53\u56de\u8f6c\u8f74\u4e3a\u7403\u4f53\u7684\u5207\u7ebf\u65f6\uff0c I=7/5mr²\u3002r\u4e3a\u7403\u4f53\u534a\u5f84\u3002
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绛旓細鐩存帴鐢鍏紡:L=Jw锛屽叾涓璍鏄氨鏄墍姹傚垰浣撶殑瑙掑姩閲忥紝J鏄垰浣撳杞酱鐨杞姩鎯噺锛寃鏄浆鍔ㄨ閫熷害銆傚湪缁忓吀鍔涘涓紝杞姩鎯噺锛堝張绉拌川閲忔儻鎬х煩锛岀畝绉版儻璺濓級閫氬父浠 鎴朖琛ㄧず锛孲I 鍗曚綅涓 kg路m²銆傚浜庝竴涓川鐐癸紝I = mr²锛屽叾涓 m 鏄叾璐ㄩ噺锛宺 鏄川鐐瑰拰杞酱鐨勫瀭鐩磋窛绂汇傝浆鍔ㄦ儻閲忓湪鏃嬭浆鍔ㄥ姏瀛...
绛旓細I=mr²銆杞姩鎯噺璁＄畻鍏紡锛欼=mr²銆傚湪缁忓吀鍔涘涓紝杞姩鎯噺锛堝張绉拌川閲忔儻鎬х煩锛岀畝绉版儻璺濓級閫氬父浠鎴朖琛ㄧず锛孲I鍗曚綅涓簁g路m²銆傚浜庝竴涓川鐐癸紝I=mr²锛屽叾涓璵鏄叾璐ㄩ噺锛宺鏄川鐐瑰拰杞酱鐨勫瀭鐩磋窛绂汇傝浆鍔ㄦ儻閲忚绠楀叕寮忥細1銆佸浜庣粏鏉嗭細褰撳洖杞酱杩囨潌鐨勪腑鐐癸紙璐ㄥ績锛夊苟鍨傜洿浜庢潌鏃禝=...
绛旓細杞姩鎯噺璁＄畻鍏紡锛1銆佸浜庣粏鏉嗭細褰撳洖杞酱杩囨潌鐨勪腑鐐癸紙璐ㄥ績锛夊苟鍨傜洿浜庢潌鏃禝=mL²/I²锛涘叾涓璵鏄潌鐨勮川閲忥紝L鏄潌鐨勯暱搴︺傚綋鍥炶浆杞磋繃鏉嗙殑绔偣骞跺瀭鐩翠簬鏉嗘椂I=mL²/3锛涘叾涓璵鏄潌鐨勮川閲忥紝L鏄潌鐨勯暱搴︺2銆佸浜庡渾鏌变綋锛氬綋鍥炶浆杞存槸鍦嗘煴浣撹酱绾挎椂I=mr²/2锛涘叾涓璵鏄渾鏌变綋鐨...
绛旓細I=mr²銆杞姩鎯噺璁＄畻鍏紡锛欼=mr²銆傚湪缁忓吀鍔涘涓紝杞姩鎯噺锛堝張绉拌川閲忔儻鎬х煩锛岀畝绉版儻璺濓級閫氬父浠鎴朖琛ㄧず锛孲I鍗曚綅涓簁g路m²銆傚浜庝竴涓川鐐癸紝I=mr²锛屽叾涓璵鏄叾璐ㄩ噺锛宺鏄川鐐瑰拰杞酱鐨勫瀭鐩磋窛绂汇傝浆鍔ㄦ儻閲忚绠楀叕寮忥細1銆佸浜庣粏鏉嗭細褰撳洖杞酱杩囨潌鐨勪腑鐐癸紙璐ㄥ績锛夊苟鍨傜洿浜庢潌鏃禝=...
绛旓細1銆杞姩鎯噺鐨勮绠鍏紡涓猴細瀵逛簬缁嗘潌锛1锛夊綋鍥炶浆杞磋繃鏉嗙殑涓偣(璐ㄥ績)骞跺瀭鐩翠簬鏉嗘椂锛屽叾涓璵鏄潌鐨勮川閲忥紝L鏄潌鐨勯暱搴︼細锛2锛夊綋鍥炶浆杞磋繃鏉嗙殑绔偣骞跺瀭鐩翠簬鏉嗘椂锛屽叾涓璵鏄潌鐨勮川閲忥紝L鏄潌鐨勯暱搴︼細2銆佸浜庡渾鏌变綋褰撳洖杞酱鏄渾鏌变綋杞寸嚎鏃讹紝鍏朵腑m鏄渾鏌变綋鐨勮川閲忥紝r鏄渾鏌变綋鐨勫崐寰勶細3銆佸浜庣粏鍦嗙幆褰撳洖杞酱...
绛旓細3銆佸疄蹇冪悆 瀵逛簬璐ㄩ噺涓簃銆佸崐寰勪负R鐨勫疄蹇冪悆浣擄紝缁曞叾鐩村緞杞寸殑杞姩鎯噺涓猴細I=(2/5)mR^2銆瀹炲績鐞冧綋鐨璐ㄩ噺鍒嗗竷鍦ㄦ墍鏈夊崐寰勬柟鍚戜笂鍧囧寑锛屽洜姝ゅ叾杞姩鎯噺鍙笌鐞冧綋鐨勮川閲忓拰鍗婂緞鏈夊叧銆傛牴鎹悆浣撶殑瀵圭О鎬э紝杞姩鎯噺鐨勮绠鍏紡鍙互绠鍖栦负涓婅堪褰㈠紡銆傚父瑙佺殑娴嬮噺杞姩鎯噺鐨勬柟娉 1銆佹棆杞硶锛氳繖鏄渶甯歌鐨勬柟娉曚箣涓銆傞氳繃...
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绛旓細涔嬪墠鏄粈涔堬紵涔︾殑涔嬪墠锛熶綘瑕佸厛鎼炴竻妤歮鏄粈涔堬紝m鏄川閲忥紝璐ㄩ噺=瀵嗗害涔樹互浣撶Н锛岃繖鐨勫瘑搴﹀氨鏄鐞冪殑鏁翠綋瀵嗗害,濡傛灉鎯崇敤闈㈠瘑搴︾畻涔熷彲浠 浣嗘鏃剁殑璐ㄩ噺灏辨槸闈㈠瘑搴︿箻浠ラ潰绉紝涓婇潰鐨勫紡瀛愬氨瑕佹敼鍐欍傚啀璇村疄闄呴棶棰橀噷闈㈠瘑搴︽槸涓嶅ソ娴嬬殑 鍥犳鎸変綋瀵嗗害绠楋紝浠涓虹洿寰勭殑鐞僲=4/3蟺r^3蟻 ,褰撳崐寰勫閲忎负dr鏃讹紝鍙煡绂荤悆...
