Question

7£®ÈçͼËùʾ£¬Á½¸ö°ëÔ²ÖùA¡¢B½ô¿¿×ž²ÖÃÓÚˮƽµØÃæÉÏ£¬ÆäÉÏÓÐÒ»¹â»¬Ô²ÖùC£¬ÈýÕ߰뾶¾ùΪR£®CµÄÖÊÁ¿Îªm£¬A¡¢BµÄÖÊÁ¿¶¼Îª$\frac{m}{2}$£¬ÓëµØÃæµÄ¶¯Ä¦²ÁÒòÊý¾ùΪ¦Ì£®ÏÖÓÃˮƽÏòÓÒµÄÁ¦À­A£¬Ê¹A»ºÂýÒƶ¯£¬Ö±ÖÁCÇ¡ºÃ½µµ½µØÃ森Õû¸ö¹ý³ÌÖÐB±£³Ö¾²Ö¹£®Éè×î´ó¾²Ä¦²ÁÁ¦µÈÓÚ»¬¶¯Ä¦²ÁÁ¦£¬ÖØÁ¦¼ÓËÙ¶ÈΪg£®Çó£º
£¨1£©Î´À­Aʱ£¬CÊܵ½B×÷ÓÃÁ¦µÄ´óСF£»
£¨2£©¶¯Ä¦²ÁÒòÊýµÄ×îСֵ¦Ì_min£»
£¨3£©AÒƶ¯µÄÕû¸ö¹ý³ÌÖУ¬À­Á¦×öµÄ¹¦W£®

Answer 1

·ÖÎö £¨1£©¸ù¾Ý¹²µãÁ¦µÄƽºâÌõ¼þÇó½âCÊܵ½B×÷ÓÃÁ¦µÄ´óСF£»
£¨2£©Ïȸù¾Ý¹²µãÁ¦Æ½ºâÌõ¼þÇó½âBÊܵ½Cˮƽ·½Ïò×î´óѹÁ¦£¬ÔÙÇó³öB¶ÔµØÃæµÄѹÁ¦£¬¸ù¾ÝĦ²ÁÁ¦µÄ¼ÆË㹫ʽÇó½â£»
£¨3£©¸ù¾Ý¶¯Äܶ¨ÀíÇó½âAÒƶ¯µÄÕû¸ö¹ý³ÌÖУ¬À­Á¦×öµÄ¹¦W£®

½â´ð ½â£º£¨1£©CÊÜÁ¦Æ½ºâ£¬ÈçͼËùʾ£º

¸ù¾ÝƽºâÌõ¼þ¿ÉµÃ£º2Fcos30¡ã=mg£¬
½âµÃCÊܵ½B×÷ÓÃÁ¦µÄ´óСΪ£ºF=$\frac{\sqrt{3}}{3}mg$£»
£¨2£©CÇ¡ºÃ½µÂäµ½µØÃæʱ£¬B¶ÔCÖ§³ÖÁ¦×î´óΪF_m£¬ÈçͼËùʾ£¬

Ôò¸ù¾ÝÁ¦µÄƽºâ¿ÉµÃ£º2F_mcos60¡ã=mg£¬
½âµÃ£ºF_m=mg£»
ËùÒÔ×î´ó¾²Ä¦²ÁÁ¦ÖÁÉÙΪ£ºf_m=F_mcos30¡ã=$\frac{\sqrt{3}}{2}mg$£¬
B¶ÔµÄµØÃæµÄѹÁ¦Îª£ºF_N=m_Bg+$\frac{1}{2}{m}_{C}g$=mg£¬
BÊܵØÃæµÄĦ²ÁÁ¦Îª£ºf=¦Ìmg£¬
¸ù¾ÝÌâÒâÓУºf_m=f£¬
½âµÃ£º¦Ì=$\frac{\sqrt{3}}{2}$£¬
ËùÒÔ¶¯Ä¦²ÁÒòÊýµÄ×îСֵΪ£º¦Ì_min=$\frac{\sqrt{3}}{2}$£»
£¨3£©CϽµµÄ¸ß¶ÈΪ£ºh=$£¨\sqrt{3}-1£©R$£¬
AµÄλÒÆΪ£ºx=$2£¨\sqrt{3}-1£©R$£¬
Ħ²ÁÁ¦×ö¹¦µÄ´óСΪ£ºW_f=fx=2$£¨\sqrt{3}-1£©¦ÌmgR$£¬
¸ù¾Ý¶¯Äܶ¨ÀíÓУºW-W_f+mgh=0£¬
½âµÃ£ºW=$£¨2¦Ì-1£©£¨\sqrt{3}-1£©mgR$£®
´ð£º£¨1£©Î´À­Aʱ£¬CÊܵ½B×÷ÓÃÁ¦µÄ´óСΪ$\frac{\sqrt{3}}{3}mg$£»
£¨2£©¶¯Ä¦²ÁÒòÊýµÄ×îСֵΪ$\frac{\sqrt{3}}{2}$£»
£¨3£©AÒƶ¯µÄÕû¸ö¹ý³ÌÖУ¬À­Á¦×öµÄ¹¦WΪ$£¨2¦Ì-1£©£¨\sqrt{3}-1£©mgR$£®

µãÆÀ ±¾ÌâÖ÷ÒªÊÇ¿¼²éÁ˹²µãÁ¦µÄƽºâºÍ¶¯Äܶ¨Àí£¬½â´ð´ËÀàÎÊÌâµÄÒ»°ã²½ÖèÊÇ£ºÈ·¶¨Ñо¿¶ÔÏó¡¢½øÐÐÊÜÁ¦·ÖÎö¡¢ÀûÓÃƽÐÐËıßÐη¨Ôò½øÐÐÁ¦µÄºÏ³É»òÕßÊÇÕý½»·Ö½â·¨½øÐÐÁ¦µÄ·Ö½â£¬È»ºóÔÚ×ø±êÖáÉϽ¨Á¢Æ½ºâ·½³Ì½øÐнâ´ð£®ÔËÓö¯Äܶ¨Àí½âÌâʱ£¬Ê×ÏÈҪѡȡÑо¿¹ý³Ì£¬È»ºó·ÖÎöÔÚÕâ¸öÔ˶¯¹ý³ÌÖÐÄÄЩÁ¦×öÕý¹¦¡¢ÄÄЩÁ¦×ö¸º¹¦£¬³õÄ©¶¯ÄÜΪ¶àÉÙ£¬¸ù¾Ý¶¯Äܶ¨ÀíÁз½³Ì½â´ð£®