Question

5£®Ò»ÇáÉþͨ¹ýÎÞĦ²ÁµÄ¶¨»¬ÂÖÓë¹Ì¶¨ÔÚˮƽµØÃæÉÏÇã½ÇΪ30¡ãµÄ¹â»¬Ð±ÃæÉϵÄÎïÌåm₁=4mÁ¬½Ó£¬ÁíÒ»¶ËºÍÌ×Ôڹ̶¨ÔÚµØÃæÉϵÄÊúÖ±¹â»¬¸ËÉϵÄÎïÌåm₂=mÁ¬½Ó£®ÒÑÖª¶¨»¬ÂÖµ½¸ËµÄ¾àÀëΪ$\sqrt{3}$L£¨ABˮƽÁ¬Ïß³¤£©£¬ÎïÌåm₂Óɾ²Ö¹´ÓABÁ¬ÏßΪˮƽµÄλÖÿªÊ¼Ï»¬£®²»¼Æ»¬ÂÖ¡¢ÎïÌåµÄ´óС£¬ÖØÁ¦¼ÓËÙ¶ÈΪg£®ÊÔÇó£º
£¨1£©m₂ÔÚÏ»¬¹ý³ÌÖеÄ×î´óËٶȣ®
£¨2£©m₂ÑØÊúÖ±¸ËÏ»¬µ½µÄ×î´ó¾àÀëʱ»úеÄÜ£¨È¡BµãËùÔÚµÄˮƽÃæΪ²Î¿¼Æ½Ã棩£®

Answer 1

·ÖÎö £¨1£©ÒÔÁ½¸öÎïÌå×é³ÉµÄϵͳΪÑо¿¶ÔÏó£¬Ö»ÓÐÖØÁ¦¶Ôϵͳ×ö¹¦£¬Æä»úеÄÜÊغ㣮µ±m₁ºÍm₂ÊÜÁ¦Ç¡ºÃƽºâʱ£¬ËÙ¶È×î´ó£¬ÓÉƽºâÌõ¼þºÍ»úеÄÜÊغ㶨ÂÉÇó½â×î´óËٶȣ®
£¨2£©µ±Á½¸öÎïÌåµÄËٶȵ½´ïÁãʱ£¬m₂Ï»¬µÄ¾àÀë×î´ó£¬ÔÙÓÉ»úеÄÜÊغ㶨ÂÉÇó½â×î´ó¾àÀ룮

½â´ð ½â£º£¨1£©Èçͼ£¬ÔÚÏ»¬¹ý³ÌÖе±m₂´¦ÓÚƽºâ״̬ʱËÙ¶È×î´ó£¬Éè×î´óËÙ¶ÈΪv₂£¬´Ëʱm₁µÄËÙ¶ÈÉèΪv₁£®
¸ù¾ÝÁ¦Æ½ºâÌõ¼þ¿ÉµÃµ½Í¼ÖеĦÈ=30¡ã
Ôòm₂ϽµµÄ¾àÀëΪh₂=L          ¢Ù
m₂ÉÏÉýµÄ¸ß¶ÈΪh₁=£¨2-$\sqrt{3}$£©Lsin30¡ã ¢Ú
¸ù¾Ý»úеÄÜÊغ㶨ÂɵÃ
  m₂gh₂-m₁gh₁=$\frac{1}{2}{m}_{1}{v}_{1}^{2}$+$\frac{1}{2}{m}_{2}{v}_{2}^{2}$   ¢Û
¸ù¾ÝÔ˶¯µÄºÏ³ÉÓë·Ö½â¿ÉµÃ v₁=v₂cos60¡ã£¬¼´ v₂=2v₁ ¢Ü
½âÉÏÊö¸÷ʽµÃ×î´óËÙ¶Èv₂=$\sqrt{£¨2\sqrt{3}-3£©gL}$ ¢Ý
£¨2£©Éèm₂ϽµµÄ×î´ó¾àÀëΪH₂£¬
Ôòm₁ÉÏÉýÊǾàÀëΪH₁=£¨$\sqrt{{H}_{2}^{2}+£¨\sqrt{3}L£©^{2}}$-$\sqrt{3}$L£©sin30¡ã  ¢Þ
¸ù¾Ý»úеÄÜÊغ㶨ÂɵÃ
 m₂gH₂=m₁gH₁                       ¢ß
m₂Ï»¬µ½×î´ó´¦µÄ»úеÄÜΪ E=-m₂gH₂     ¢à
½â¢Þ¢ß¢àµÃ E=-$\frac{4\sqrt{3}}{3}$mgL  ¢á
´ð£º
£¨1£©m₂ÔÚÏ»¬¹ý³ÌÖеÄ×î´óËÙ¶ÈÊÇ$\sqrt{£¨2\sqrt{3}-3£©gL}$£®
£¨2£©m₂ÑØÊúÖ±¸ËÏ»¬µ½µÄ×î´ó¾àÀëʱ»úеÄÜÊÇ-$\frac{4\sqrt{3}}{3}$mgL£®

µãÆÀ ±¾ÌâÊÇÉþϵµÄϵͳ»úеÄÜÊغãÎÊÌ⣬ƽºâÖ»ÊÇÆäÖеÄÒ»¸ö״̬£¬ÓÉƽºâÌõ¼þºÍ»úеÄÜÊغ㶨Âɼ°¼¸ºÎ֪ʶ½áºÏÇó½â£®