14.
随着人们生活水平的提高,冬季滑雪成为人们休闲娱乐的一项主要运动,某滑雪场的滑道示意图如图所示。半径
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3ER%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmtext%3Em%3C%2Fmtext%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的光滑
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmfrac%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmn%3E4%3C%2Fmn%3E%3C%2Fmfrac%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
圆弧轨道ABC与足够长的粗糙轨道CD在C处平滑连接,O为圆弧轨道ABC的圆心,B点为圆弧轨道的最低点,半径OA、OC与竖直线OB的夹角分别为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmn%3E53%3C%2Fmn%3E%3Cmo%3E%C2%B0%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
和
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmn%3E37%3C%2Fmn%3E%3Cmo%3E%C2%B0%3C%2Fmo%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
。质量为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3Em%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E30%3C%2Fmn%3E%3Cmtext%3Ekg%3C%2Fmtext%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
的小孩(视为质点)从A点左侧高为
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3Eh%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E0.8%3C%2Fmn%3E%3Cmtext%3Em%3C%2Fmtext%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
处的P点水平滑出,恰从A点沿切线方向进入圆弧轨道。已知小孩与轨道CD间的动摩擦因数
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3E%CE%BC%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E0.8%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
, 最大静摩擦力等于滑动摩擦力,重力加速度
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3Eg%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E10%3C%2Fmn%3E%3Cmsup%3E%3Cmrow%3E%3Cmtext%3Em%2Fs%3C%2Fmtext%3E%3C%2Fmrow%3E%3Cmtext%3E2%3C%2Fmtext%3E%3C%2Fmsup%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
,
![](//math.21cnjy.com/MathMLToImage?mml=%3Cmath+xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmrow%3E%3Cmi%3Esin%3C%2Fmi%3E%3Cmn%3E37%3C%2Fmn%3E%3Cmo%3E%C2%B0%3C%2Fmo%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E0.6%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmi%3Ecos%3C%2Fmi%3E%3Cmn%3E37%3C%2Fmn%3E%3Cmo%3E%C2%B0%3C%2Fmo%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E0.8%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmath%3E)
, 不计空气阻力。求:
(1)小孩水平滑出时的初速度大小
;
(2)小孩经过B点时,对圆弧轨道的压力大小;
(3)小孩在轨道CD上滑行的距离。(计算结果保留三位有效数字)
![](//tikupic.21cnjy.com/ct20241o/f8/8e/f88e8e27acc3209c4f6aa04363905e4b.png)