Question

5£®ÈçͼËùʾ£¬Ò»°ë¾¶ÎªRµÄÔ²±íʾһÖùÐÎÇøÓòµÄºá½ØÃ棬ԲÐÄ×ø±êΪ£¨0£¬R£©£¬ÔÚÖùÐÎÇøÓòÄÚ¼ÓÒ»·½Ïò´¹Ö±ÓÚÖ½ÃæÏòÀï´Å¸ÐӦǿ¶ÈΪBµÄÔÈÇ¿´Å³¡£¬Ôڴų¡ÓÒ²àÓÐһƽÐÐÓÚxÖá·ÅÖõÄƽÐаåµçÈÝÆ÷£¬Á½°å¼ä¾àºÍ°å³¤¾ùΪ2R£¬N°åÓëxÖáÖغÏÇҽӵأ®Ò»ÖÊÁ¿Îªm¡¢µçºÉÁ¿Îª-qµÄ´øµçÁ£×Ó£¬ÓÉ×ø±êÔ­µãOÒÔÏàͬËÙÂÊ¡¢²»Í¬·½ÏòÑØÖ½ÃæÉäÈëµÚÒ»ÏóÏÞºó£¬ÔÚÉä³ö´Å³¡Ê±Á£×ÓµÄËٶȶ¼Æ½ÐÐÓÚxÖᣮ²»¼ÆÖØÁ¦£®ÊÔÇó£º
£¨1£©´øµç΢Á£Ôڴų¡ÖÐÔ˶¯µÄËٶȴóС£¿
£¨2£©Èô´øµçÁ£×Ó´ÓOµãÉäÈë´Å³¡Ê±µÄËÙ¶ÈÇ¡ÓëxÖá³É¦È=60¡ã½Ç£¬Ôò¸ÃÁ£×ÓÔڴų¡ÖÐÔ˶¯µÄʱ¼äΪ¶àÉÙ£¿Éä³ö´Å³¡Ê±µÄλÖÃ×ø±êΪ¶àÉÙ£¿
£¨3£©Èôʹ£¨2£©ÖеĴøµçÁ£×ÓÄܹ»´ÓƽÐаåµçÈÝÆ÷µÄÓÒ¶ËÉä³ö£¬M°åµÄµçÊÆ·¶Î§Îª¶à´ó£¿£¨È¡N°åµÄµçÊÆΪÁãµçÊÆ£®£©

Answer 1

·ÖÎö £¨1£©ÓÉÓÚÁ£×Ó´ÓÔ²Ðδų¡ÇøÓòÉä³öºó¾ùÑØˮƽ·½ÏòÉä³ö£¬Óɼ¸ºÎ¹Øϵ֪µÀ£¬Á£×Ó×öÔÈËÙÔ²ÖÜÔ˶¯µÄ°ë¾¶Óë´Å³¡Ô²µÄ°ë¾¶Ïàͬ£®ÔÙÓÉÂåÂØ×ÈÁ¦ÌṩÏòÐÄÁ¦¿ÉÒÔÇóµÃÁ£×ÓÉä³öµÄËٶȣ®
£¨2£©Èô´øµçÁ£×Ó´ÓOµãÉäÈë´Å³¡Ê±µÄËÙ¶ÈÇ¡ÓëxÖá³É¦È=60¡ã½Ç£¬ÔòÁ£×Ó¾­¹ý´Å³¡Æ«×ªºó±äΪˮƽ£¬Æ«×ª½ÇΪ60¡ã£¬ÓÉÖÜÆÚ¹«Ê½¾ÍÄÜÇó³öÔڴų¡µÄʱ¼ä£®À뿪´Å³¡ºó×öÀàƽÅ×Ô˶¯£¬ÔÙÓÉÀàƽÅ׵ĹæÂɾÍͨ±¨Çó³ö´òÖÐλÖÃ×ø±ê£®
£¨3£©ÓÉÓÚûÓиæËßMN°åµçÊƵĸߵͣ¬ËùÒÔÁ£×ÓˮƽÉä³öºó¿ÉÄÜÏòÉÏ»òÏòÏÂ×öÀàƽÅ×Ô˶¯£¬ÕâÑù£¬ÓÉÀàƽÅ׵ĹæÂÉÇó³öÁ½ÖÖÇé¿öϵļÓËٶȣ¬´Ó¶øÄÜÇó³öÁ½°åµÄµçÊƲ´Ó¶ø¾ÍÄÜÇó³öM°åµÄµçÊÆ·¶Î§£®

½â´ð ½â£º£¨1£©ÈçͼËùʾ£¬Óɼ¸ºÎ¹Øϵ¿ÉÖª£¬¹ì¼£Ô²µÄÔ²ÐÄO¡¯¡¢³öÉäµãPÓëO₁ºÍO¹¹³ÉÁâÐΣ¬Òò´Ë
  Á£×ÓÔڴų¡ÖÐÔ˶¯µÄ¹ì¼£°ë¾¶ÎªR£¬Ôò    
  $qBv=m\frac{{v}^{2}}{R}$  
  ËùÒÔ $v=\frac{qBR}{m}$    
£¨2£©$t=\frac{¦ÈR}{v}=\frac{¦Ðm}{3qB}$     ÈçͼËùʾ¡ä¡ä£¬Éä³ö´Å³¡µÄλÖÃ×ø±êΪ£º
  $x=Rsin¦È=\frac{\sqrt{3}}{2}R$ 
  $y=Rcos¦È=\frac{1}{2}R$  
£¨3£©ÉèÁ£×ÓÔÚÁ½°å¼äÔ˶¯µÄʱ¼äΪt£¬Ôò
  $t=\frac{2R}{v}=\frac{2m}{qB}$  
  ÉèÁ£×Ó´Ó°åÓÒ²àÉä³öʱ²àÏòλÒƵĴóСΪd£¬Ôò
  $d=\frac{1}{2}a{t}^{2}$       
  $a=\frac{F}{m}=\frac{Eq}{m}$    
  ËùÒÔ  $E=\frac{q{B}^{2}d}{2m}$     
  ËùÒÔ£¬µ±Á£×Ó´ÓN°åµÄÓÒ±ßÔµÉä³öʱ£¬d₁=$\frac{1}{2}R$Ôò
    ${U}_{NM1}={E}_{1}•2R=\frac{q{B}^{2}{R}^{2}}{2m}$     
  ËùÒÔ ${¦Õ}_{M1}={¦Õ}_{N}-{U}_{M1}=-\frac{q{B}^{2}{R}^{2}}{2m}$  
  µ±Á£×Ó´ÓM°åµÄÓÒ±ßÔµÉä³öʱ£¬d₂=$\frac{3}{2}R$Ôò
  ${U}_{NM2}={E}_{2}•2R=\frac{3{q}^{2}{B}^{2}}{2m}$   
  ËùÒÔ  ${¦Õ}_{M2}={U}_{M2N}+{¦Õ}_{N}=\frac{3q{B}^{2}{R}^{2}}{2m}$  
  ËùÒÔ£¬Á£×ÓÄܹ»´ÓƽÐаåµçÈÝÆ÷µÄÓÒ¶ËÉä³ö£¬M°åµÄµçÊÆ·¶Î§Îª£º
  $-\frac{q{B}^{2}{R}^{2}}{2m}¡Ü{¦Õ}_{M}¡Ü\frac{3q{B}^{2}{R}^{2}}{2m}$
´ð£º£¨1£©´øµç΢Á£Ôڴų¡ÖÐÔ˶¯µÄËٶȴóСΪ$\frac{qBR}{m}$£®
£¨2£©Èô´øµçÁ£×Ó´ÓOµãÉäÈë´Å³¡Ê±µÄËÙ¶ÈÇ¡ÓëxÖá³É¦È=60¡ã½Ç£¬Ôò¸ÃÁ£×ÓÔڴų¡ÖÐÔ˶¯µÄʱ¼äΪ$\frac{¦Ðm}{3qB}$£¬Éä³ö´Å³¡Ê±µÄλÖÃ×ø±êΪ£¨$\frac{\sqrt{3}}{2}R$£¬$\frac{1}{2}R$£©£®
£¨3£©Èôʹ£¨2£©ÖеĴøµçÁ£×ÓÄܹ»´ÓƽÐаåµçÈÝÆ÷µÄÓÒ¶ËÉä³ö£¬M°åµÄµçÊÆ·¶Î§Îª$-\frac{q{B}^{2}{R}^{2}}{2m}¡Ü{¦Õ}_{M}¡Ü\frac{3q{B}^{2}{R}^{2}}{2m}$£®

µãÆÀ ±¾ÌâµÄÇÉÃîÔÚÓÚ£º¢ÙËùÓÐÁ£×Ó¾ùˮƽÉä³öÔ²Ðδų¡ÇøÓò£¬ÓÉ´Ë¿ÉÒÔÍƶÏÁ£×Ó×öÔÈËÙÔ²ÖÜÔ˶¯µÄ°ë¾¶Óë´Å³¡Ô²°ë¾¶ÏàµÈ£»¢ÚÔÚµÚ¶þÎʵķ½ÏòÉä³öµÄÁ£×ÓÔڵ糡ÖÐ×öÀàƽÅ×Ô˶¯£¬ÓÉÓÚûÓиæËßMNÁ½°åµçÊƵĸߵͣ¬ËùÒÔÁ£×Ó¿ÉÄÜÏòÉÏ»òÏòÏÂ×öÀàƽÅ×Ô˶¯£¬ÓÉÀàƽÅ×¹æÂÉÇó³öµçÊƲ´Ó¶øÇó³öM°åµÄµçÊÆ£®