Question

7£®½«Ò»ÖÊÁ¿ÎªmµÄ¿ÉÊÓΪÖʵãµÄСÇòÓÃÒ»³¤ÎªlµÄϸÉþË©½Ó£¬Ï¸ÉþµÄÁíÒ»¶Ë¹Ì¶¨ÔÚÌ컨°åÉϵÄOµã£¬ÔÚOµãÓÒÏ·½µÄAλÖö¤Ò»¹â»¬µÄ¶¤×Ó£¬ÒÑÖªAOÁ½µãµÄ¸ß¶È²îΪ$\frac{l}{2}$£®ÏÖ½«Ð¡Çò´ÓͼʾλÖÃÎÞ³õËÙ¶ÈÊÍ·Å£¬´ËλÖÃÓëOµãµÈ¸ßÇÒ´ËʱϸÉþ¸ÕºÃ±Á½ô£¬µ±Ï¸ÉþÓëAµãµÄ¶¤×ÓÏàÅöºó£¬Ð¡Çò½«ÔÚÊúÖ±ÃæÄÚÈƶ¤×Ó×öÔ²ÖÜÔ˶¯£®²»¼ÆϸÉþÓ붤×ÓÅöײʱµÄ»úеÄÜËðʧ£®
£¨1£©Èç¹ûAµãµ½¹ýOµãÊúÖ±ÏߵľàÀëΪ$\frac{\sqrt{5}}{4}$lʱ£¬Ð¡Çò¸ÕºÃÔ˶¯µ½×îµÍµãʱϸÉþ¶ÏÁÑ£¬ÔòϸÉþÄܳÐÊܵÄ×î´óÀ­Á¦ÓëÖØÁ¦µÄ±ÈÖµÊǶàÉÙ£¿
£¨2£©Îª±£Ö¤Ï¸ÉþÔÚ²»¶ÏÁѵÄÇé¿öÏ£¬Ð¡ÇòÄÜÔÚÊúÖ±ÃæÄÚ×öÍêÕûµÄÔ²ÖÜÔ˶¯£¬ÔòAµãµ½OµãÊúÖ±ÏߵľàÀ뷶ΧÊǶàÉÙ£¿

Answer 1

·ÖÎö £¨1£©¸ù¾Ý¼¸ºÎ֪ʶÇó³öСÇòÈƶ¤×Ó×öÔ²ÖÜÔ˶¯µÄ°ë¾¶£¬ÓÉ»úеÄÜÊغ㶨ÂÉÇó³öСÇòÔ˶¯µ½×îµÍµãʱµÄËٶȣ¬ÔÙÓÉÅ£¶ÙµÚ¶þ¶¨ÂɺÍÏòÐÄÁ¦¹«Ê½Çó½âϸÉþÄܳÐÊܵÄ×î´óÀ­Á¦ÓëÖØÁ¦µÄ±ÈÖµ£®
£¨2£©Ð¡ÇòÇ¡ºÃ×öÍêÕûµÄÔ²ÖÜÔ˶¯Ê±£¬ÔÚÔ²ÖܵÄ×î¸ßµãÓÉÖØÁ¦ÌṩÏòÐÄÁ¦£¬ÓÉ´ËÁÐʽ£¬½áºÏ»úеÄÜÊغ㶨ÂÉÇó³öAµãµ½OµãÊúÖ±ÏߵľàÀë×î´óÖµ£®µ±Ï¸Éþ¸ÕÒª¶ÏÁÑʱ£¬ÔÚ×îµÍµãÀ­Á¦´ïµ½×î´óÖµ£¬ÓÉÅ£¶ÙµÚ¶þ¶¨ÂɺͻúеÄÜÊغ㶨ÂÉÇó³öAµãµ½OµãÊúÖ±ÏߵľàÀë×îСֵ£¬´Ó¶øµÃµ½Aµãµ½OµãÊúÖ±ÏߵľàÀ뷶Χ£®

½â´ð ½â£º£¨1£©ÉèСÇòÈƶ¤×Ó×öÔ²ÖÜÔ˶¯µÄ°ë¾¶Îªr£®¸ù¾Ý¼¸ºÎ֪ʶµÃ£º
r=l-$\sqrt{£¨\frac{l}{2}£©^{2}+£¨\frac{\sqrt{5}}{4}l£©^{2}}$=$\frac{1}{4}$l
СÇò´Ó¿ªÊ¼µ½Ô˶¯µ½×îµÍµãµÄ¹ý³Ì£¬ÓÉ»úеÄÜÊغ㶨Âɵãº
mg£¨r+$\frac{l}{2}$£©=$\frac{1}{2}m{v}^{2}$
ÔÚ×îµÍµã£¬ÓÉÅ£¶ÙµÚ¶þ¶¨Âɵãº
T_m-mg=m$\frac{{v}^{2}}{r}$
ÁªÁ¢½âµÃ£ºÏ¸ÉþÄܳÐÊܵÄ×î´óÀ­Á¦ÓëÖØÁ¦Ö®±È $\frac{{T}_{m}}{mg}$=7
£¨2£©ÉèСÇòÇ¡ºÃ×öÍêÕûµÄÔ²ÖÜÔ˶¯Ê±£¬Aµãµ½OµãÊúÖ±ÏߵľàÀëΪs₁£®ÔÚÔ²ÖܵÄ×î¸ßµãÓÉÖØÁ¦ÌṩÏòÐÄÁ¦£¬ÔòÓУº
mg=m$\frac{{v}_{1}^{2}}{l-\sqrt{£¨\frac{l}{2}£©^{2}+{s}_{1}^{2}}}$
´Ó¿ªÊ¼µ½Ô²ÖܵÄ×î¸ßµã£¬ÓÉ»úеÄÜÊغ㶨Âɵãº
mg{$\frac{l}{2}$-[l-$\sqrt{£¨\frac{l}{2}£©^{2}+{s}_{1}^{2}}$]}=$\frac{1}{2}m{v}_{1}^{2}$
ÁªÁ¢½âµÃ£ºs₁=$\frac{\sqrt{7}}{6}$l
ÉèСÇòÔÚ×îµÍµãϸÉþ¸ÕÒª¶ÏÁÑʱ£¬Aµãµ½OµãÊúÖ±ÏߵľàÀëΪs₂£®ÔÚ×îµÍµã£¬ÓÉÅ£¶ÙµÚ¶þ¶¨Âɵãº
T_m-mg=m$\frac{{v}_{2}^{2}}{l-\sqrt{£¨\frac{l}{2}£©^{2}+{s}_{2}^{2}}}$
´Ó¿ªÊ¼µ½Ô²ÖܵÄ×îµÍµã£¬ÓÉ»úеÄÜÊغ㶨Âɵãº
mg{$\frac{l}{2}$+[l-$\sqrt{£¨\frac{l}{2}£©^{2}+{s}_{2}^{2}}$]}=$\frac{1}{2}m{v}_{2}^{2}$
ÁªÁ¢½âµÃ£ºs₂=$\frac{\sqrt{5}}{4}$l
ËùÒÔAµãµ½OµãÊúÖ±ÏߵľàÀ뷶ΧÊÇ£º$\frac{\sqrt{7}}{6}$l¡Üs£¼$\frac{\sqrt{5}}{4}$l
´ð£º£¨1£©Ï¸ÉþÄܳÐÊܵÄ×î´óÀ­Á¦ÓëÖØÁ¦µÄ±ÈÖµÊÇ7£®
£¨2£©Aµãµ½OµãÊúÖ±ÏߵľàÀ뷶ΧÊÇ $\frac{\sqrt{7}}{6}$l¡Üs£¼$\frac{\sqrt{5}}{4}$l£®

µãÆÀ ¶ÔÓÚÔ²ÖÜÔ˶¯¶¯Á¦Ñ§ÎÊÌ⣬¹Ø¼üÊÇ·ÖÎöÊÜÁ¦Çé¿ö£¬Ñ°ÕÒÏòÐÄÁ¦µÄÀ´Ô´£¬ÕÒ³öÒþº¬µÄÁÙ½çÌõ¼þ£®