Question

11£®ÈçͼΪijÖÖÓã¶ü×Ô¶¯Í¶·ÅÆ÷ÖеÄͶ¶ü¹Ü×°ÖÃʾÒâͼ£¬ÆäÏ°벿ABÊÇÒ»³¤Îª2RµÄÊúֱϸ¹Ü£¬Éϰ벿BCÊǰ뾶ΪRµÄËÄ·ÖÖ®Ò»Ô²»¡Íä¹Ü£¬¹Ü¿ÚÑØˮƽ·½Ïò£¬AB¹ÜÄÚÓÐÒ»Ô­³¤ÎªR¡¢Ï¶˹̶¨µÄÇáÖʵ¯»É£®Í¶¶üʱ£¬Ã¿´Î×ܽ«µ¯»É³¤¶ÈѹËõµ½0.5RºóËø¶¨£¬ÔÚµ¯»ÉÉ϶ηÅÖÃÒ»Á£Óã¶ü£¬½â³ýËø¶¨£¬µ¯»É¿É½«Óã¶üµ¯Éä³öÈ¥£®ÉèÖÊÁ¿ÎªmµÄÓã¶üµ½´ï¹Ü¿ÚCʱ£¬¶Ô¹Ü±ÚµÄ×÷ÓÃÁ¦Ç¡ºÃΪÁ㣮²»¼ÆÓã¶üÔÚÔ˶¯¹ý³ÌÖеĻúеÄÜËðʧ£¬ÇÒËø¶¨ºÍ½â³ýËø¶¨Ê±£¬¾ù²»¸Ä±äµ¯»ÉµÄµ¯ÐÔÊÆÄÜ£®ÒÑÖªÖØÁ¦¼ÓËÙ¶ÈΪg£®Çó£º
£¨1£©ÖÊÁ¿ÎªmµÄÓã¶üµ½´ï¹Ü¿ÚCʱµÄËٶȴóСv₁£»
£¨2£©µ¯»ÉѹËõµ½0.5RʱµÄµ¯ÐÔÊÆÄÜE_P£»
£¨3£©ÒÑÖªµØÃæÓëË®ÃæÏà¾à1.5R£¬Èôʹ¸ÃͶ¶ü¹ÜÈÆAB¹ÜµÄÖÐÖáÏßOO^-ÔÚ90¡ã½ÇµÄ·¶Î§ÄÚÀ´»Ø»ºÂýת¶¯£¬Ã¿´Îµ¯Éäʱֻ·ÅÖÃÒ»Á£Óã¶ü£¬Óã¶üµÄÖÊÁ¿ÔÚ$\frac{2}{3}m$µ½mÖ®¼ä±ä»¯£¬ÇÒ¾ùÄÜÂ䵽ˮÃ森³ÖÐøͶ·Å×ã¹»³¤Ê±¼äºó£¬Óã¶üÄܹ»Â䵽ˮÃæµÄ×î´óÃæ»ýSÊǶàÉÙ£¿

Answer 1

·ÖÎö £¨1£©Óã¶üµ½´ï¹Ü¿ÚCʱ×öÔ²ÖÜÔ˶¯µÄÏòÐÄÁ¦ÍêÈ«ÓÉÖØÁ¦Ìṩ£¬ÓÐÅ£¶ÙµÚ¶þ¶¨ÂÉÁгöÏòÐÄÁ¦µÄ·½³Ì£¬ËٶȿÉÇó£®
£¨2£©²»¼ÆÓã¶üÔÚÔ˶¯¹ý³ÌÖеĻúеÄÜËðʧ£¬ËùÒÔÓã¶üÔö¼ÓµÄ»úеÄܶ¼Êǵ¯»É×ö¹¦µÄ½á¹û£¬Óɹ¦ÄܹØϵ֪µÀµ¯»É¾ßÓеĵ¯ÐÔÊÆÄܵÈÓÚÓã¶üÔö¼ÓµÄ»úеÄÜ£®
£¨3£©·Ö±ðÇó³öÖÊÁ¿ÊÇmºÍ$\frac{2}{3}m$µÄÓã¶üÀ뿪CʱµÄËٶȣ¬ÔÙÀûÓÃƽÅ×Ô˶¯¹æÂÉÇó³öÂ䵽ˮƽÃ浽תÖáÖ®¼äµÄ¾àÀ룬תÖáת¹ý90¡ãʱ£¬Óã¶üÔÚˮƽÃæÐγÉÁ½¸ö$\frac{1}{4}$Ô²Ãæ»ý£¬Öмä¼ÐµÄ»·ÐÎÃæ»ý¼´ÎªËùÇó£®

½â´ð ½â£º£¨1£©ÔÚCµã£¬ÓÉÅ£¶ÙµÚ¶þ¶¨ÂÉ£º$mg=m\frac{{{v_1}^2}}{R}$
¿ÉµÃ  ${v_1}=\sqrt{gR}$
£¨2£©´Óµ¯»ÉÊͷŵ½×î¸ßµãCµÄ¹ý³ÌÖУ¬¸ù¾Ý¶¯Äܶ¨ÀíÓУº
 ${W_F}+{W_G}=\frac{1}{2}m{v_1}^2-0$
ÔòµÃµ¯»ÉµÄµ¯Á¦×ö¹¦ ${W_F}-mg£¨2.5R£©=\frac{1}{2}m{£¨\sqrt{gR}£©^2}-0$
½âµÃ W_F=3mgR
¹Êµ¯»ÉѹËõµ½0.5RʱµÄµ¯ÐÔÊÆÄÜ E_P=W_F=3mgR
£¨3£©Óã¶üÀ뿪¹Ü¿ÚCºó×öƽÅ×Ô˶¯£¬Æ½Å×Ô˶¯µÄʱ¼äΪ $t=\sqrt{\frac{2h}{g}}=\sqrt{\frac{2¡Á4.5R}{g}}=3\sqrt{\frac{R}{g}}$
Óã¶üµÄÖÊÁ¿Îªmʱ£¬Ë®Æ½Éä³ÌΪ£ºx₁=v₁t=3R
Óã¶üµÄÖÊÁ¿Îª$\frac{2}{3}m$ʱ£¬¸ù¾Ý¶¯Äܶ¨ÀíµÃ
  W_F-$\frac{2}{3}mg$•£¨2.5R£©=$\frac{1}{2}£¨\frac{2}{3}m£©v{¡ä}^{2}$-0
½âµÃ v¡ä=2$\sqrt{gR}$
ˮƽÉä³Ì x₂=v¡ät=6R
r₁=x₁+r=4R£¬r₂=x₂+r=7R
¹ÊÓã¶üÄܹ»Â䵽ˮÃæµÄ×î´óÃæ»ý $s=\frac{1}{4}£¨¦Ð{r_2}^2-¦Ð{r_1}^2£©$  
½âµÃ s=8.25¦ÐR²£®
´ð£º£¨1£©ÖÊÁ¿ÎªmµÄÓã¶üµ½´ï¹Ü¿ÚCʱµÄËٶȴóСv₁ÊÇ$\sqrt{gR}$£®
£¨2£©µ¯»ÉѹËõµ½0.5RʱµÄµ¯ÐÔÊÆÄÜE_PÊÇ3mgR£®
£¨3£©Óã¶üÄܹ»Â䵽ˮÃæµÄ×î´óÃæ»ýSÊÇ8.25¦ÐR²£®

µãÆÀ ±¾Ì⿼²éÁËÔ²ÖÜÔ˶¯×î¸ßµãµÄ¶¯Á¦Ñ§·½³ÌºÍƽÅ×Ô˶¯¹æÂÉ£¬×ªÖáת¹ý90¡ãÓã¶üÔÚˮƽÃæÉÏÐγÉÔ²ÖÜÊǽâ¾öÎÊÌâµÄ¹Ø¼ü£®