Question

(08年蘇州中學(xué)期中) (12分)質(zhì)量為m=1kg的小物塊輕輕放在水平勻速運(yùn)動(dòng)的傳送帶上的P點(diǎn)，隨傳送帶運(yùn)動(dòng)到A點(diǎn)后水平拋出，小物塊恰好無(wú)碰撞的沿圓弧切線從B點(diǎn)進(jìn)入豎直光滑圓孤軌道下滑。B、C為圓弧的兩端點(diǎn)，其連線水平。已知圓弧半徑R=1.0m圓弧對(duì)應(yīng)圓心角θ=106°，軌道最低點(diǎn)為O，A點(diǎn)距水平面的高度h=0.8m。小物塊離開C點(diǎn)后恰能無(wú)碰撞的沿固定斜面向上運(yùn)動(dòng)，0.8s后經(jīng)過D點(diǎn)，物塊與斜面間的動(dòng)摩擦因數(shù)為μ₁=0.33（g=10m/s²，sin37°=0.6，cos37°=0.8）試求：

（1）小物塊離開A點(diǎn)的水平初速度v₁。

（2）小物塊經(jīng)過O點(diǎn)時(shí)對(duì)軌道的壓力。

（3）斜面上CD間的距離。

（4）假設(shè)小物塊與傳送帶間的動(dòng)摩擦因數(shù)為μ₂=0.3，傳送帶的速度為5m/s，則PA間的距離？

Answer 1

解析：（1）對(duì)小物塊，由A到B有

 

                                                                                                      

在B點(diǎn)

                                                                                                      

所以        

                                                                      

（2）對(duì)小物塊，由B到O有

 

                                                                                                      

其中

                                                                                                        

在O點(diǎn)

                                                                                                        

所以N=43N

 

由牛頓第三定律知對(duì)軌道的壓力為

                                                                                                        

（3）物塊沿斜面上滑：

                                                                                                        

所以

 

物塊沿斜面下滑：

                                                                                                        

由機(jī)械能守恒知

 

小物塊由C上升到最高點(diǎn)歷時(shí)

                                                                                                        

小物塊由最高點(diǎn)回到D點(diǎn)歷時(shí)

                                                                                                        

故

                                                                                                        

即

                                                                                                        

（4）小物塊在傳送帶上加速過程：

                                                                                                      

       PA間的距離是