Question

1£®Èçͼ£¬Æøµæµ¼¹ìÉÏ»¬¿éµÄÖÊÁ¿ÎªM£¬¹³ÂëµÄÖÊÁ¿Îªm£¬ÕÚ¹âÌõ¿í¶ÈΪd£¬Á½¹âµçÃżäµÄ¾àÀëΪL£¬ÆøÔ´¿ªÍ¨ºó»¬¿éÔÚÇ£ÒýÁ¦µÄ×÷ÓÃÏÂÏȺóͨ¹ýÁ½¸ö¹âµçÃŵÄʱ¼äΪ¡÷t₁ºÍ¡÷t₂£®µ±µØµÄÖØÁ¦¼ÓËÙ¶ÈΪg
£¨1£©ÓÃÉÏÊö×°ÖòâÁ¿»¬¿é¼ÓËٶȵıí´ïʽΪ$a=\frac{d^2}{2L}£¨\frac{1}{{¡÷t{\;}_2^2}}-\frac{1}{¡÷t_1^2}£©$£¨ÓÃÒÑÖªÁ¿±íʾ£©£»
£¨2£©ÓÃÉÏÊö×°ÖÃ̽¾¿»¬¿é¼ÓËÙ¶ÈaÓëÖÊÁ¿M¼°À­Á¦FµÄ¹Øϵ£¬ÒªÊ¹ÉþÖÐÀ­Á¦½üËƵÈÓÚ¹³ÂëµÄÖØÁ¦£¬ÔòmÓëMÖ®¼äµÄ¹ØϵӦÂú×ãM£¾£¾m£»
£¨3£©ÓÃÉÏÊö×°ÖÃ̽¾¿ÏµÍ³ÔÚÔ˶¯ÖеĻúеÄܹØϵ£¬»¬¿é´Ó¹âµçÃÅ1Ô˶¯µ½¹âµçÃÅ2µÄ¹ý³ÌÖÐÂú×ã¹Øϵʽ$mgL=\frac{1}{2}£¨M+m£©£¨\frac{d^2}{¡÷t_2^2}-\frac{d^2}{¡÷t_1^2}£©$ʱ£¨ÓÃÒÑÖªÁ¿±íʾ£©£¬ÏµÍ³»úеÄÜÊغ㣮Èô²âÁ¿¹ý³ÌÖз¢ÏÖϵͳ¶¯ÄÜÔöÁ¿×ÜÊÇ´óÓÚ¹³ÂëÖØÁ¦ÊÆÄܵļõÉÙÁ¿£¬¿ÉÄܵÄÔ­ÒòÊǵ¼¹ì²»Æ½£¬ÓҶ˸ߣ®

Answer 1

·ÖÎö £¨1£©¹âµçÃŲâÁ¿»¬¿é˲ʱËٶȵÄÔ­ÀíÊÇÕÚ¹âÌõͨ¹ý¹âµçÃŵÄËٶȿÉÒÔÓÃƽ¾ùËٶȴúÌæ¼´v=$\frac{d}{v}$£¬ÔÙ¸ù¾ÝÔ˶¯Ñ§¹«Ê½¼´¿ÉÇó³öÎïÌåµÄ¼ÓËÙ¶Èa£®
£¨2£©Ö»ÓÐÔÚ»¬¿éÖÊÁ¿Ô¶´óÓÚ¹³ÂëÖÊÁ¿Ê±£¬²Å¿É½üËÆÈÏΪ»¬¿éÊܵ½µÄÀ­Á¦µÈÓÚ¹³ÂëµÄÖØÁ¦£®
£¨3£©ÊµÑéÔ­ÀíÊÇ£ºÇó³öͨ¹ý¹âµçÃÅ1ʱµÄËÙ¶Èv₁£¬Í¨¹ý¹âµçÃÅ1ʱµÄËÙ¶Èv₂£¬²â³öÁ½¹âµçÃżäµÄ¾àÀëA£¬ÔÚÕâ¸ö¹ý³ÌÖУ¬¼õÉÙµÄÖØÁ¦ÊÆÄÜÄÜ£º¡÷E_p=mgL£¬Ôö¼ÓµÄ¶¯ÄÜΪ£º¡÷E_k=$\frac{1}{2}$£¨M+m£©v₂²-$\frac{1}{2}$£¨M+m£©v₁²£»ÔٱȽϼõÉÙµÄÖØÁ¦ÊÆÄÜÓëÔö¼ÓµÄ¶¯ÄÜÖ®¼äµÄ¹Øϵ£®
½«Æøµæµ¼¹ìÇãбºó£¬ÓÉÓÚ»¬¿éµÄÖØÁ¦ÊÆÄܵÄÔö¼Ó»ò¼õÉÙûÓмÇÈ룬¹ÊÔö¼ÓµÄ¶¯ÄܺͼõÉÙµÄÖØÁ¦ÊÆÄܲ»ÏàµÈ£¬Á½ÖÖÇé¿ö·Ö±ðÌÖÂÛ£®

½â´ð ½â£º£¨1£©¸ù¾ÝÕÚ¹â°åͨ¹ý¹âµçÃŵÄËٶȿÉÒÔÓÃƽ¾ùËٶȴúÌæµÃ£º
»¬¿éͨ¹ýµÚÒ»¸ö¹âµçÃÅʱµÄËٶȣº${v_1}=\frac{d}{{¡÷{t_1}}}$
»¬¿éͨ¹ýµÚ¶þ¸ö¹âµçÃÅʱµÄËٶȣº${v_2}=\frac{d}{{¡÷{t_2}}}$
»¬¿éµÄ¼ÓËٶȣº$a=\frac{{v}_{2}-{v}_{1}}{¡÷t}=\frac{{d}^{2}}{2L}£¨\frac{1}{¡÷t{\;}_{2}^{2}}-\frac{1}{¡÷{t}_{1}^{2}}£©$
£¨2£©µ±»¬¿éÖÊÁ¿MÔ¶´óÓÚ¹³ÂëÖÊÁ¿m ʱ£¬Ï¸ÏßÖÐÀ­Á¦½üËƵÈÓÚ¹³ÂëÖØÁ¦£®
£¨3£©ÊµÑéÔ­ÀíÊÇ£ºÇó³öͨ¹ý¹âµçÃÅ1ʱµÄËÙ¶Èv₁£¬Í¨¹ý¹âµçÃÅ1ʱµÄËÙ¶Èv₂£¬²â³öÁ½¹âµçÃżäµÄ¾àÀëA£¬ÔÚÕâ¸ö¹ý³ÌÖУ¬¼õÉÙµÄÖØÁ¦ÊÆÄÜÄÜ£º¡÷Ep=mgL£¬Ôö¼ÓµÄ¶¯ÄÜΪ£º
$\frac{1}{2}$£¨M+m£©v₂²-$\frac{1}{2}$£¨M+m£©v₁²
ÎÒÃÇÑéÖ¤µÄÊÇ£º¡÷E_pÓë¡÷E_kµÄ¹Øϵ£¬¼´ÑéÖ¤£º¡÷E_p=¡÷E_k´úÈëµÃ£º
mgL=$\frac{1}{2}$£¨M+m£©v₂²-$\frac{1}{2}$£¨M+m£©v₁²
¼´$mgL=\frac{1}{2}£¨M+m£©£¨\frac{d^2}{¡÷t_2^2}-\frac{d^2}{¡÷t_1^2}£©$
½«Æøµæµ¼¹ìÇãбºó£¬ÓÉÓÚ»¬¿éµÄÖØÁ¦ÊÆÄܵÄÔö¼Ó»ò¼õÉÙûÓмÇÈ룬¹ÊÔö¼ÓµÄ¶¯ÄܺͼõÉÙµÄÖØÁ¦ÊÆÄܲ»ÏàµÈ£ºÈô×ó²à¸ß£¬ÏµÍ³¶¯ÄÜÔö¼ÓÁ¿´óÓÚÖØÁ¦ÊÆÄܼõÉÙÁ¿£»ÈôÓÒ²à¸ß£¬ÏµÍ³¶¯ÄÜÔö¼ÓÁ¿Ð¡ÓÚÖØÁ¦ÊÆÄܼõÉÙÁ¿£»¹ÊÆøµæµ¼¹ìµÄ×ó²àÆ«¸ß£®
¹Ê´ð°¸Îª£º
£¨1£©$a=\frac{d^2}{2L}£¨\frac{1}{{¡÷t{\;}_2^2}}-\frac{1}{¡÷t_1^2}£©$
£¨2£©M£¾£¾m    
£¨3£©$mgL=\frac{1}{2}£¨M+m£©£¨\frac{d^2}{¡÷t_2^2}-\frac{d^2}{¡÷t_1^2}£©$     µ¼¹ì²»Æ½£¬ÓҶ˸ß

µãÆÀ ÕýÈ·½â´ðʵÑéÎÊÌâµÄÇ°ÌáÊÇÃ÷ȷʵÑéÔ­Àí£¬´ÓʵÑéÔ­Àí³ö·¢½øÐзÖÎöËùÐèʵÑéÆ÷²Ä¡¢ÊµÑé²½Öè¡¢Ëù²âÊý¾ÝµÈ£¬»áÆðµ½Ê°빦±¶µÄЧ¹û£®