Question

4£®½«Ò»¸ù³¤ÎªLµÄ¹â»¬Ï¸¸ÖË¿ABCDEÖƳÉÈçͼËùʾµÄÐÎ×´£¬²¢¹Ì¶¨ÔÚÊúֱƽÃæÄÚ£®ÆäÖÐAD¶ÎÊúÖ±£¬DE¶ÎΪ$\frac{3}{4}$Ô²»¡£¬Ô²ÐÄΪO£¬EΪԲ»¡×î¸ßµã£¬CÓëE¡¢DÓëO·Ö±ðµÈ¸ß£¬BC=$\frac{1}{4}$AC£®½«ÖÊÁ¿ÎªmµÄСÖéÌ×ÔÚ¸ÖË¿ÉÏÓɾ²Ö¹ÊÍ·Å£¬²»¼Æ¿ÕÆø×èÁ¦£¬ÖØÁ¦¼ÓËÙ¶ÈΪg£®
£¨1£©Ð¡ÖéÓÉCµãÊÍ·Å£¬Çóµ½´ïEµãµÄËٶȴóСv₁£»
£¨2£©Ð¡ÖéÓÉBµãÊÍ·Å£¬´ÓEµã»¬³öºóÇ¡ºÃײµ½Dµã£¬ÇóÔ²»¡µÄ°ë¾¶R£»
£¨3£©ÓûʹСÖéµ½´ïEµãÓë¸ÖË¿¼äµÄµ¯Á¦³¬¹ý$\frac{mg}{4}$£¬ÇóÊÍ·ÅСÖéµÄλÖ÷¶Î§£®

Answer 1

·ÖÎö £¨1£©Ð¡ÖéÓÉCµãÊͷźó£¬Ð¡ÇòµÄ»úеÄÜÊغ㣬ÓÉ´ËÇó³öСÇòµ½´ïEµãµÄËٶȴóСv₁£»
£¨2£©Ð¡ÖéÓÉBµãÊÍ·Å£¬´ÓEµã»¬³öºó×öƽÅ×Ô˶¯£¬½«Ô˶¯·Ö½â¼´¿ÉÇó³öÔ²»¡µÄ°ë¾¶R£»
£¨3£©¶ÔСÖé¼õСÊÜÁ¦·ÖÎö£¬ÓÉ»úеÄÜÊغ㣬½áºÏÅ£¶ÙµÚ¶þ¶¨Âɼ´¿ÉÇó³öÊÍ·ÅСÖéµÄλÖ÷¶Î§£®

½â´ð ½â£º£¨1£©Ñ¡È¡OµãΪ0ÊÆÄܵ㣬CµãÓëEµãÏà¶ÔÓÚOµÄ¸ß¶È¶¼ÊÇR£»Ð¡ÖéÓÉCµãÊͷźó£¬Ð¡ÇòµÄ»úеÄÜÊغ㣬Ôò£º
$mgR=mgR+\frac{1}{2}m{v}_{1}^{2}$
ËùÒÔ£ºv₁=0£»
 £¨2£©Ð¡Öé´ÓEµã»¬³öºó×öƽÅ×Ô˶¯£¬ÊúÖ±·½Ïò£ºR=$\frac{1}{2}g{t}^{2}$
ËùÒÔ£ºt=$\sqrt{\frac{2R}{g}}$
ˮƽ·½Ïò£ºR=v₂t
ËùÒÔ£º${v}_{2}=\frac{R}{t}=\sqrt{\frac{gR}{2}}$
BÏà¶ÔÓÚOµãµÄ¸ß¶È£ºh_B=R+$\frac{1}{4}\overline{AC}$
ÆäÖУº$\overline{AC}=L-\frac{3}{4}•2¦ÐR-R$
Bµ½EµÄ¹ý³ÌÖУº$mg{h}_{B}=mgR+\frac{1}{2}m{v}_{2}^{2}$
ÁªÁ¢¿ÉµÃ£º$R=\frac{2L}{4+3¦Ð}$£»
£¨3£©ÉèÊͷŵ㠵ĸ߶ÈÏà¶ÔÓÚCµãΪh£¬ÈôСÖé×Óµ½´ïEʱ£¬Óë¸ÖË¿¼äµÄµ¯Á¦³¬¹ý$\frac{mg}{4}$ÓÐÁ½ÖÖÇé¿ö£º
¢ñ¡¢Ð¡ÖéÊܵ½µÄµ¯Á¦·½ÏòÏòÉÏ£¬Ôò£º$mg-\frac{1}{4}mg=\frac{m{v}_{3}^{2}}{R}$
ÓÉ»úеÄÜÊغãµÃ£º$mg£¨h+R£©=mgR+\frac{1}{2}m{v}_{3}^{2}$
ÁªÁ¢µÃ£ºh=$\frac{3L}{4£¨4+3¦Ð£©}$
¢ò¡¢Ð¡ÖéÊܵ½µÄµ¯Á¦·½ÏòÏòÏ£¬Ôò£º$mg+\frac{1}{4}mg=\frac{m{v}_{4}^{2}}{R}$
ÓÉ»úеÄÜÊغãµÃ£º$mg£¨h¡ä+R£©=mgR+\frac{1}{2}m{v}_{4}^{2}$
ÁªÁ¢µÃ£ºh¡ä=$\frac{5L}{4£¨4+3¦Ð£©}$
¼´µ±Ð¡Öé×Ó´ÓCµãÉÏ·½µÍÓÚ$\frac{3L}{4£¨4+3¦Ð£©}$´¦»¬Ï»ò¸ßÓÚ$\frac{5L}{4£¨4+3¦Ð£©}$´¦»¬ÏÂʱ£¬Ð¡Öéµ½´ïEµãÓë¸ÖË¿¼äµÄµ¯Á¦³¬¹ý$\frac{mg}{4}$£®
´ð£º£¨1£©Ð¡ÖéÓÉCµãÊÍ·Å£¬µ½´ïEµãµÄËٶȴóСΪ0£»
£¨2£©Ð¡ÖéÓÉBµãÊÍ·Å£¬´ÓEµã»¬³öºóÇ¡ºÃײµ½Dµã£¬Ô²»¡µÄ°ë¾¶RΪ$\frac{2L}{4+3¦Ð}$£»
£¨3£©ÓûʹСÖéµ½´ïEµãÓë¸ÖË¿¼äµÄµ¯Á¦³¬¹ý$\frac{mg}{4}$£¬ÔòÐèҪСÖé×Ó´ÓCµãÉÏ·½µÍÓÚ$\frac{3L}{4£¨4+3¦Ð£©}$´¦»¬Ï»ò¸ßÓÚ$\frac{5L}{4£¨4+3¦Ð£©}$´¦»¬Ï£®

µãÆÀ ¸ÃÌâ½áºÏƽÅ×Ô˶¯ÒÔ¼°ÊúֱƽÃæÄÚµÄÔ²ÖÜÔ˶¯¿¼²é»úеÄÜÊغ㣬ÊôÓÚ»úеÄÜÊغ㶨ÂɵÄÓ¦ÓÃÖÐÄѶȴóµÄÌâÄ¿£¬ÔÚ½â´ðµÄ¹ý³ÌÖÐҪעÒâ¶Ô¹ý³ÌµÄ·ÖÎö£¬Í¬Ê±Òª×¢Òâ¸÷֪ʶµãµÄ½áºÏ£®