Question

如图3－2－16所示，在高出水平地面h＝1.8 m的光滑平台上放置一质量M＝2 kg、由两种不同材料连接成一体的薄板A，其右段长度l₁＝0.2 m且表面光滑，左段表面粗糙．在A最右端放有可视为质点的物块B，其质量m＝1 kg，B与A左段间动摩擦因数μ＝0.4.开始时二者均静止，现对A施加F＝20 N水平向右的恒力，待B脱离A(A尚未露出平台)后，将A取走．B离开平台后的落地点与平台右边缘的水平距离x＝1.2 m．(取g＝10 m/s²)求：

图3－2－16

(1)B离开平台时的速度v_B.

(2)B从开始运动到刚脱离A时，B运动的时间t_B和位移x_B.

(3)A左段的长度l₂.

Answer 1

解析　(1)设物块B平抛运动的时间为t，由运动学知识可得

h＝gt²                                                                                                                                                                                                                                     ①

x＝v_Bt                                                                                                                       ②

联立①②式，代入数据得

v_B＝2 m/s                                                                                                                 ③

(2)设B的加速度为a_B，由牛顿第二定律和运动学的知识得

μmg＝ma_B                                                                                                                                                                                                                                 ④

v_B＝a_Bt_B                                                                                                                                                                                                                                       ⑤

x_B＝a_Bt_B²                                                                                                                                                                                                                                           ⑥

联立③④⑤⑥式，代入数据得

t_B＝0.5 s                                                                                                                   ⑦

x_B＝0.5 m                                                                                                                 ⑧

(3)设B刚开始运动时A的速度为v₁，由动能定理得

Fl₁＝Mv₁²                                                                                                                                                                                                                         ⑨

设B运动后A的加速度为a_A，由牛顿第二定律和运动学的知识得

F－μmg＝Ma_A                                                                                                                                                                                                                     ⑩

l₂＋x_B＝v₁t_B＋a_At_B²                                                                                                                                                                                                             ⑪

联立⑦⑧⑨⑩⑪式，代入数据得

l₂＝1.5 m                                                                                                                  ⑫

答案　(1)2 m/s　(2)0.5 s　0.5 m　(3)1.5 m