Question

如圖3－2－16所示，在高出水平地面h＝1.8 m的光滑平臺(tái)上放置一質(zhì)量M＝2 kg、由兩種不同材料連接成一體的薄板A，其右段長(zhǎng)度l₁＝0.2 m且表面光滑，左段表面粗糙．在A最右端放有可視為質(zhì)點(diǎn)的物塊B，其質(zhì)量m＝1 kg，B與A左段間動(dòng)摩擦因數(shù)μ＝0.4.開始時(shí)二者均靜止，現(xiàn)對(duì)A施加F＝20 N水平向右的恒力，待B脫離A(A尚未露出平臺(tái))后，將A取走．B離開平臺(tái)后的落地點(diǎn)與平臺(tái)右邊緣的水平距離x＝1.2 m．(取g＝10 m/s²)求：

圖3－2－16

(1)B離開平臺(tái)時(shí)的速度v_B.

(2)B從開始運(yùn)動(dòng)到剛脫離A時(shí)，B運(yùn)動(dòng)的時(shí)間t_B和位移x_B.

(3)A左段的長(zhǎng)度l₂.

Answer 1

解析　(1)設(shè)物塊B平拋運(yùn)動(dòng)的時(shí)間為t，由運(yùn)動(dòng)學(xué)知識(shí)可得

h＝gt²                                                                                                                                                                                                                                     ①

x＝v_Bt                                                                                                                       ②

聯(lián)立①②式，代入數(shù)據(jù)得

v_B＝2 m/s                                                                                                                 ③

(2)設(shè)B的加速度為a_B，由牛頓第二定律和運(yùn)動(dòng)學(xué)的知識(shí)得

μmg＝ma_B                                                                                                                                                                                                                                 ④

v_B＝a_Bt_B                                                                                                                                                                                                                                       ⑤

x_B＝a_Bt_B²                                                                                                                                                                                                                                           ⑥

聯(lián)立③④⑤⑥式，代入數(shù)據(jù)得

t_B＝0.5 s                                                                                                                   ⑦

x_B＝0.5 m                                                                                                                 ⑧

(3)設(shè)B剛開始運(yùn)動(dòng)時(shí)A的速度為v₁，由動(dòng)能定理得

Fl₁＝Mv₁²                                                                                                                                                                                                                         ⑨

設(shè)B運(yùn)動(dòng)后A的加速度為a_A，由牛頓第二定律和運(yùn)動(dòng)學(xué)的知識(shí)得

F－μmg＝Ma_A                                                                                                                                                                                                                     ⑩

l₂＋x_B＝v₁t_B＋a_At_B²                                                                                                                                                                                                             ⑪

聯(lián)立⑦⑧⑨⑩⑪式，代入數(shù)據(jù)得

l₂＝1.5 m                                                                                                                  ⑫

答案　(1)2 m/s　(2)0.5 s　0.5 m　(3)1.5 m