Question

1£®ÈçͼËùʾ£¬°ë¾¶ÎªRµÄ°ëÇòÐÎÌÕ¹Þ£¬¹Ì¶¨ÔÚ¿ÉÒÔÈÆÊúÖ±ÖáÐýתµÄˮƽת̨ÉÏ£¬×ªÌ¨×ªÖáÓë¹ýÌÕ¹ÞÇòÐÄOµÄ¶Ô³ÆÖáOO¡äÖغϣ®×ªÌ¨ÒÔÒ»¶¨½ÇËٶȦØÔÈËÙת¶¯£®Ò»ÖÊÁ¿ÎªmµÄСÎï¿éÂäÈëÌÕ¹ÞÄÚ£¬¾­¹ýÒ»¶Îʱ¼äºó£¬Ð¡Îï¿éËæÌÕ¹ÞÒ»Æðת¶¯ÇÒÏà¶Ô¹Þ±Ú¾²Ö¹£¬ËüºÍOµãµÄÁ¬ÏßÓëOO¡äÖ®¼äµÄ¼Ð½Ç¦ÈΪ60¡ã£®ÖØÁ¦¼ÓËٶȴóСΪg£®
£¨1£©Èô¦Ø=¦Ø₀£¬Ð¡Îï¿éÊܵ½µÄĦ²ÁÁ¦Ç¡ºÃΪÁ㣬Çó¦Ø₀£»
£¨2£©¦Ø=£¨1¡Àk£©¦Ø₀£¬ÇÒ0£¼k£¼£¼1£¬ÇóСÎï¿éÊܵ½µÄĦ²ÁÁ¦´óСºÍ·½Ïò£®

Answer 1

·ÖÎö £¨1£©Èô¦Ø=¦Ø₀£¬Ð¡Îï¿éÊܵ½µÄĦ²ÁÁ¦Ç¡ºÃΪÁ㣬¿¿ÖØÁ¦ºÍÖ§³ÖÁ¦µÄºÏÁ¦ÌṩÏòÐÄÁ¦£¬¸ù¾ÝÅ£¶ÙµÚ¶þ¶¨ÂÉÇó³ö½ÇËٶȵĴóС£®
£¨2£©µ±¦Ø£¾¦Ø₀£¬ÖØÁ¦ºÍÖ§³ÖÁ¦µÄºÏÁ¦²»¹»ÌṩÏòÐÄÁ¦£¬Ä¦²ÁÁ¦·½ÏòÑعޱÚÇÐÏßÏòÏ£¬¸ù¾ÝÅ£¶ÙµÚ¶þ¶¨ÂÉÇó³öĦ²ÁÁ¦µÄ´óС£®µ±¦Ø£¼¦Ø₀£¬ÖØÁ¦ºÍÖ§³ÖÁ¦µÄºÏÁ¦´óÓÚÏòÐÄÁ¦£¬ÔòĦ²ÁÁ¦µÄ·½ÏòÑعޱÚÇÐÏßÏòÉÏ£¬¸ù¾ÝÅ£¶ÙµÚ¶þ¶¨ÂÉÇó³öĦ²ÁÁ¦µÄ´óС£®

½â´ð ½â£º£¨1£©µ±Ä¦²ÁÁ¦ÎªÁ㣬֧³ÖÁ¦ºÍÖØÁ¦µÄºÏÁ¦ÌṩÏòÐÄÁ¦£¬ÓУº
mgtan¦È=mRsin¦È¦Ø₀²£¬½âµÃ
${¦Ø}_{0}=\sqrt{\frac{2g}{R}}$
£¨2£©µ±¦Ø=£¨1+k£©¦Ø₀ʱ£¬ÖØÁ¦ºÍÖ§³ÖÁ¦µÄºÏÁ¦²»¹»ÌṩÏòÐÄÁ¦£¬Ä¦²ÁÁ¦·½ÏòÑعޱÚÇÐÏßÏòÏ£¬
¸ù¾ÝÅ£¶ÙµÚ¶þ¶¨Âɵã¬fcos60¡ã+Ncos30¡ã=mRsin60¡ã¦Ø²£®
fsin60¡ã+mg=Nsin30¡ã
ÁªÁ¢Á½Ê½½âµÃ$f=\frac{\sqrt{3}k£¨2+k£©}{2}mg$

µ±¦Ø=£¨1-k£©¦Ø₀ʱ£¬Ä¦²ÁÁ¦·½ÏòÑعޱÚÇÐÏßÏòÉÏ£¬
¸ù¾ÝÅ£¶ÙµÚ¶þ¶¨Âɵã¬Ncos30¡ã-fcos60¡ã=mRsin60¡ã¦Ø²£®
mg=Nsin30¡ã+fsin60¡ã
ÁªÁ¢Á½Ê½½âµÃf=$\frac{\sqrt{3}k£¨2-k£©}{2}mg$£®
´ð£º£¨1£©Ð¡Îï¿éÊܵ½µÄĦ²ÁÁ¦Ç¡ºÃΪÁãʱ£¬${¦Ø}_{0}=\sqrt{\frac{2g}{R}}$£®
£¨2£©µ±¦Ø=£¨1+k£©¦Ø₀ʱ£¬Ä¦²ÁÁ¦·½ÏòÑعޱÚÇÐÏßÏòÏ£¬´óСΪ$f=\frac{\sqrt{3}k£¨2+k£©}{2}mg$£®
µ±¦Ø=£¨1-k£©¦Ø₀ʱ£¬Ä¦²ÁÁ¦·½ÏòÑعޱÚÇÐÏßÏòÉÏ£¬´óСΪ$\frac{\sqrt{3}k£¨2-k£©}{2}mg$£®

µãÆÀ ½â¾ö±¾ÌâµÄ¹Ø¼ü¸ãÇåÎï¿é×öÔ²ÖÜÔ˶¯ÏòÐÄÁ¦µÄÀ´Ô´£¬½áºÏÅ£¶ÙµÚ¶þ¶¨ÂÉ£¬×¥×¡ÊúÖ±·½ÏòÉϺÏÁ¦ÎªÁ㣬ˮƽ·½ÏòÉϵĺÏÁ¦ÌṩÏòÐÄÁ¦½øÐÐÇó½â£®