Question

16£®ÈçͼËùʾ£¬ÊúÖ±¹Ì¶¨·ÅÖõĴֲÚбÃæABµÄ϶ËÓë¹â»¬µÄÔ²»¡¹ìµÀBCDÔÚBµãÏàÇУ¬Ô²»¡¹ìµÀµÄ°ë¾¶ÎªR£¬Ô²ÐÄOÓëA¡¢DÔÚͬһˮƽÃæÉÏ£¬CµãΪԲ»¡¹ìµÀ×îµÍµã£¬¡ÏCOB=¦È=30¡ã£®ÏÖʹһÖÊÁ¿ÎªmµÄСÎï¿é´ÓDµãÎÞ³õËٶȵØÊÍ·Å£¬Ð¡Îï¿éÓë´Ö²ÚбÃæAB¼äµÄ¶¯Ä¦²ÁÒòÊý¦Ì£¼tan¦È£¬Ôò¹ØÓÚСÎï¿éµÄÔ˶¯Çé¿ö£¬ÏÂÁÐ˵·¨ÕýÈ·µÄÊÇ£¨¡¡¡¡£©

• A£® СÎï¿é¿ÉÄÜÔ˶¯µ½Aµã
• B£® СÎï¿é¾­¹ý½Ï³¤Ê±¼äºó»áÍ£ÔÚCµã
• C£® СÎï¿éͨ¹ýÔ²»¡¹ìµÀ×îµÍµãCʱ£¬¶ÔCµãµÄ×î´óѹÁ¦´óСΪ3mg
• D£® СÎï¿éͨ¹ýÔ²»¡¹ìµÀ×îµÍµãCʱ£¬¶ÔCµãµÄ×îСѹÁ¦´óСΪ£¨3-$\sqrt{3}$£©mg

Answer 1

·ÖÎö Óɼ¸ºÎ֪ʶµÃÖª£¬Ð±ÃæµÄÇã½ÇµÈÓÚ30¡ã£®ÎïÌå´ÓAµãÎÞ³õËٶȻ¬Ïºó£¬ÓÉÓÚ¿Ë·þĦ²ÁÁ¦×ö¹¦£¬ÎïÌåÔÚбÃæÉÏÔ˶¯Ê±»úеÄܲ»¶Ï¼õС£¬µ½´ïµÄ×î´ó¸ß¶ÈÔ½À´Ô½Ð¡£¬×îÖÕÔÚBEÔ²»¡ÉÏÀ´»ØÔ˶¯£¬µ½´ïBµãµÄËÙ¶ÈΪÁ㣮
µ±ÎïÌåµÚÒ»´Î¾­¹ýCµãʱ£¬ËÙ¶È×î´ó£¬¶ÔCµãµÄѹÁ¦×î´ó£¬µ±×îºóÎȶ¨ºó£¬ÎïÌåÔÚBEÖ®¼äÔ˶¯Ê±£¬¾­¹ýCµãʱËÙ¶È×îС£¬ÎïÌå¶ÔCµãµÄѹÁ¦×îС£¬¸ù¾Ý¶¯Äܶ¨ÀíÇó³ö×î´óËٶȺÍ×îСËٶȣ¬ÔÙÓÉÅ£¶ÙÔ˶¯¶¨ÂÉÇó½â×î´óѹÁ¦ºÍ×îСѹÁ¦£®

½â´ð ½â£ºAB¡¢ÎïÌå´ÓDµãÎÞ³õËٶȻ¬Ïºó£¬ÓÉÓÚÔÚAB¶ÎÒª¿Ë·þĦ²ÁÁ¦×ö¹¦£¬ËùÒÔÎïÌåÔÚбÃæÉÏÔ˶¯Ê±»úеÄܲ»¶Ï¼õС£¬ÔÚбÃæÉÏÉýµÄ×î´ó¸ß¶ÈÔ½À´Ô½Ð¡£¬²»¿ÉÄÜÔ˶¯µ½Aµã£¬×îÖÕÔÚBEÔ²»¡ÉÏÀ´»ØÔ˶¯£¬²»»áÍ£ÔÚCµã£®¹ÊA¡¢B´íÎó£®
C¡¢Îï¿éµÚÒ»´ÎÔ˶¯µ½CʱËÙ¶È×î´ó£¬¶Ô¹ìµÀµÄѹÁ¦×î´ó£®Îï¿é´ÓDµÚÒ»´ÎÔ˶¯µ½C¹ý³Ì£¬Óɶ¯Äܶ¨ÀíµÃ£º
    mgR=$\frac{1}{2}m{v}_{1}^{2}$
Éè´Ëʱ¹ìµÀ¶ÔÎïÌåµÄÖ§³ÖÁ¦F₁£¬ÓÉÅ£¶ÙµÚ¶þ¶¨Âɵãº
  F₁-mg=m$\frac{{v}_{1}^{2}}{R}$ÁªÁ¢½âµÃ£ºF₁=3mg£¬ÓÉÅ£¶ÙµÚÈý¶¨ÂÉÖªÎï¿é¶ÔCµãµÄ×î´óѹÁ¦Îª3mg£®¹ÊCÕýÈ·£®
D¡¢µ±×îºóÎȶ¨ºó£¬ÎïÌåÔÚBEÖ®¼äÔ˶¯Ê±£¬ÉèÎïÌå¾­¹ýCµãµÄËÙ¶ÈΪv₂£¬Óɶ¯Äܶ¨ÀíµÃ£º
    mgR£¨1-cos¦È£©=$\frac{1}{2}m{v}_{2}^{2}$
Éè¹ìµÀ¶ÔÎïÌåµÄÖ§³ÖÁ¦F₂£¬ÓÉÅ£¶ÙµÚ¶þ¶¨Âɵãº
   F₂-mg=m$\frac{{v}_{2}^{2}}{R}$
ÁªÁ¢½âµÃ£ºF₂=3mg-2mgcos¦È=£¨3-$\sqrt{3}$£©mg
ÓÉÅ£¶ÙµÚÈý¶¨ÂÉ¿ÉÖª£¬ÎïÌå¶ÔCµãµÄ×îСѹÁ¦Îª£¨3-$\sqrt{3}$£©mg£®¹ÊDÕýÈ·£®
¹ÊÑ¡£ºCD

µãÆÀ ±¾ÌâÊǶ¯Äܶ¨ÀíÓëÅ£¶ÙÔ˶¯¶¨ÂɵÄ×ÛºÏÓ¦Ó㬹ؼüÊÇ·ÖÎöÎïÌåµÄÔ˶¯¹ý³Ì£¬Ã÷È·CµãÔ²ÖÜÔ˶¯µÄÏòÐÄÁ¦µÄÀ´Ô´£ººÏÍâÁ¦£®