Question

4£®ÌìÎÄѧ¼Ò½«Ïà¾à½Ï½ü¡¢½öÔڱ˴˵ÄÒýÁ¦×÷ÓÃÏÂÔËÐеÄÁ½¿ÅºãÐdzÆΪ˫ÐÇ£®Ë«ÐÇϵͳÔÚÒøºÓϵÖкÜÆձ飮ÀûÓÃË«ÐÇϵͳÖÐÁ½¿ÅºãÐǵÄÔ˶¯ÌØÕ÷¿ÉÒÔÍÆËã³öËüÃǵÄ×ÜÖÊÁ¿£®ÒÑ֪ij˫ÐÇϵͳÖÐÁ½¿ÅºãÐÇΧÈÆËüÃÇÁ¬ÏßÉϵÄijһ¹Ì¶¨µã·Ö±ð×öÔÈËÙÔ²ÖÜÔ˶¯£¬ÖÜÆÚ¾ùΪT£¬Á½¿ÅºãÐÇÖ®¼äµÄ¾àÀëΪr
£¨1£©ÊÔÍÆËãÕâ¸öË«ÐÇϵͳµÄ×ÜÖÊÁ¿£¨ÒýÁ¦³£Á¿ÎªG£©
£¨2£©Ñо¿·¢ÏÖ£¬Ë«ÐÇϵͳÑÝ»¯¹ý³ÌÖУ¬Á½ÐǵÄ×ÜÖÊÁ¿¡¢¾àÀëºÍÖÜÆÚ¾ù¿ÉÄÜ·¢Éú±ä»¯£®Èôij˫ÐÇϵͳÖÐÁ½ÐǾ­¹ýÒ»¶Îʱ¼äµÄÑÝ»¯ºó£¬Á½ÐÇ×ÜÖÊÁ¿±äΪԭÀ´µÄk±¶£¬Á½ÐǼä¾à±äΪԭÀ´µÄn±¶£¬Ôò´Ëʱ˫ÐÇ×öÔ²ÖÜÔ˶¯µÄÖÜÆÚ±äΪԭÀ´µÄ¶àÉÙ±¶£¿

Answer 1

·ÖÎö £¨1£©ÕâÊÇÒ»¸öË«ÐǵÄÎÊÌ⣬Á½¿ÅºãÐÇΧÈÆËüÃÇÁ¬ÏßÉϵÄijһ¹Ì¶¨µã·Ö±ð×öÔÈËÙÔ²ÖÜÔ˶¯£¬ËüÃÇÖ®¼äµÄÍòÓÐÒýÁ¦Ìṩ¸÷×ÔµÄÏòÐÄÁ¦£¬Á½¿ÅºãÐÇÓÐÏàͬµÄ½ÇËٶȺÍÖÜÆÚ£¬½áºÏÅ£¶ÙµÚ¶þ¶¨ÂɺÍÍòÓÐÒýÁ¦¶¨Âɽâ¾öÎÊÌ⣮
£¨2£©Ë«ÐÇ¿¿Ï໥¼äµÄÍòÓÐÒýÁ¦ÌṩÏòÐÄÁ¦£¬¾ßÓÐÏàͬµÄ½ÇËٶȣ¬¸ù¾ÝÅ£¶ÙµÚ¶þ¶¨ÂÉ·Ö±ð¶ÔÁ½ÐǽøÐÐÁÐʽ£¬À´Çó½â£®

½â´ð ½â£º£¨1£©ÉèÁ½¿ÅºãÐǵÄÖÊÁ¿·Ö±ðΪm₁¡¢m₂£¬×öÔ²ÖÜÔ˶¯µÄ°ë¾¶·Ö±ðΪr₁¡¢r₂£¬½ÇËٶȷֱðΪ¦Ø₁£¬¦Ø₂£®¸ù¾ÝÌâÒâÓР
¦Ø₁=¦Ø₂ ¡­¢Ù
r₁+r₂=r¡­¢Ú
¸ù¾ÝÍòÓÐÒýÁ¦¶¨ÂɺÍÅ£¶Ù¶¨ÂÉ£¬ÓУºG$\frac{{m}_{1}{m}_{2}}{{r}^{2}}$=m₁¦Ø₁²r₁¡­¢Û
 G$\frac{{m}_{1}{m}_{2}}{{r}^{2}}$=m₂¦Ø₂²r₂¡­¢Ü
ÁªÁ¢ÒÔÉϸ÷ʽ½âµÃ£ºr₁=$\frac{{m}_{2}r}{{m}_{1}+{m}_{2}}$¡­¢Ý
¸ù¾Ý½ÇËÙ¶ÈÓëÖÜÆڵĹØϵ֪£º¦Ø₁=¦Ø₂=$\frac{2¦Ð}{T}$¡­¢Þ
ÁªÁ¢¢Û¢Ý¢Þʽ½âµÃ£ºm₁+m₂=$\frac{4{¦Ð}^{2}}{{T}^{2}G}$r³£¬
£¨2£©Éèm₁µÄ¹ìµÀ°ë¾¶ÎªR₁£¬m₂µÄ¹ìµÀ°ë¾¶ÎªR₂£®Á½ÐÇÖ®¼äµÄ¾àÀëΪl£®
ÓÉÓÚËüÃÇÖ®¼äµÄ¾àÀëºã¶¨£¬Òò´ËË«ÐÇÔÚ¿Õ¼äµÄÈÆÏòÒ»¶¨Ïàͬ£¬Í¬Ê±½ÇËٶȺÍÖÜÆÚÒ²¶¼Ïàͬ£®ÓÉÏòÐÄÁ¦¹«Ê½¿ÉµÃ£º
¶Ôm₁£º$G\frac{{m}_{1}{m}_{2}}{{l}^{2}}$=${m}_{1}\frac{4{¦Ð}^{2}}{{T}^{2}}{R}_{1}$----¢ß
¶Ôm₂£º$G\frac{{m}_{1}{m}_{2}}{{l}^{2}}$=${m}_{2}\frac{4{¦Ð}^{2}}{{T}^{2}}{R}_{2}$----¢à
 ÓÖÒòΪR₁Ê®R₂=l£¬m₁+m₂=M     ¢á
Óɢߢà¢áʽ¿ÉµÃ ${T}^{2}=\frac{4{¦Ð}^{2}{l}^{3}}{G£¨{m}_{1}+{m}_{2}£©}$
ËùÒÔµ±Á½ÐÇ×ÜÖÊÁ¿±äΪKM£¬Á½ÐÇÖ®¼äµÄ¾àÀë±äΪԭÀ´µÄn±¶£¬
Ô²ÖÜÔ˶¯µÄÖÜÆÚ${T¡ä}^{2}=\frac{4{¦Ð}^{2}{£¨nl£©}^{3}}{G£¨{m}_{1}¡ä+{m}_{2}¡ä£©}$=$\frac{4{¦Ð}^{2}{n}^{3}{l}^{3}}{GkM}$=$\frac{{n}^{3}}{k}$T²
¼´T¡ä=$\sqrt{\frac{{n}^{3}}{k}}T$
´Ëʱ˫ÐÇ×öÔ²ÖÜÔ˶¯µÄÖÜÆÚ±äΪԭÀ´µÄ$\sqrt{\frac{{n}^{3}}{k}}$±¶
´ð£º£¨1£©Õâ¸öË«ÐÇϵͳµÄ×ÜÖÊÁ¿ÊÇ=$\frac{4{¦Ð}^{2}}{{T}^{2}G}$r³£»
£¨2£©´Ëʱ˫ÐÇ×öÔ²ÖÜÔ˶¯µÄÖÜÆÚ±äΪԭÀ´µÄ$\sqrt{\frac{{n}^{3}}{k}}$±¶£®

µãÆÀ ±¾ÌâÊÇË«ÐÇÎÊÌ⣬ÓëÎÀÐÇÈƵØÇòÔ˶¯Ä£ÐͲ»Í¬£¬Á½¿ÅÐǶ¼ÈÆͬһԲÐÄ×öÔÈËÙÔ²ÖÜÔ˶¯£¬¹Ø¼üץסÌõ¼þ£ºÏàͬµÄ½ÇËٶȺÍÖÜÆÚ£®