一 10
一道立体模型的摩擦力题
高三|〖苦海泛舟〗 qiusir 2021
![](http://www.qiusir.com/wp-content/gallery/physics/2021-01-10.png)
两根长圆柱轴完全相同、轴线在同一面内,且平面与水平面夹角为θ=30°。现将一个大的匀质圆柱体放置两轴之间,且大圆柱体轴线分别与两长圆柱体轴的轴线形成的两个平面夹角为α=60°。试求解以下问题:
一、若大圆柱体恰好静止在轴上,可认为滑动摩擦力与最大静摩擦力相等,则大圆柱体与轴之间的动摩擦因数μ应满足什么条件?
二、已知某种材质的大圆柱体与轴的动摩擦因数μ=2。某一时刻,左侧轴做逆时针转动,同时右侧轴做顺时针转动,两轴转动时边缘处的线速度大小均为v=3m/s,则由于大圆柱体受力发生变化造成其将沿轴下滑,求大圆柱体下滑稳定时的速度。
一:
![2F_Ncos{\frac{\alpha}{2}}=mgcos{\theta}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_8c87ce9fbcb27c6d6b162d6e1a4dd697.gif)
![2F_f=mgsin{\theta}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_6d2be448f4832c2bbc43cb329fe020f5.gif)
![F_f\leq\mu F_N](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_1666cc2abd4c5f035fd7fb93e8e7d4b1.gif)
![\mu\geq0.5](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_d15e4f431240e87df8917b07834e3e7c.gif)
二:
![2f_1sin{\frac{\alpha}{2}}+2F_Ncos{\frac{\alpha}{2}}=mgcos{\theta}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_23fc0e5bc07703afacd7b348eca963fd.gif)
![2f_2=mgsin{\theta}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_efa66c5ba66e8a1068fbeef28027d89b.gif)
![\sqrt{{f_1}^2+{f_2}^2}=\mu F_N](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_c356e613d3f077134d89ad50c8df269b.gif)
![45\frac{f_2}{f_1}+\frac{f_1}{f_2}=16\sqrt{3}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_6ccb2aca3cefb6c9a22fee16f5af7ee9.gif)
得
![\frac{f_2}{f_1}=\frac{\sqrt{3}}{3}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_0c5d29a0f4c004924a0a952880753a73.gif)
![\frac{1}{15\sqrt{3}}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_f803eb71c0025a1b1913fecd2b49f67c.gif)
![\frac{v_1}{v_2}=\frac{f_1}{f_2}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_2e1efeb7fc463a1551aff0d78877de64.gif)
![v_2=\sqrt{3}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_8892544825625454c880d0d5ed0c3f72.gif)
![\frac{\sqrt{3}}{15}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_adf398548f58958b67da7233b8980495.gif)
对于上面的解,洪诚天同学检测到第二个结果对应的
![F_N](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_cb0072665108e047bc0c03b5bd3a35ef.gif)
如果直接引入两个摩擦力的夹角
![\beta](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_b0603860fcffe94e5b8eec59ed813421.gif)
代入
![f_1=\mu F_Nsin\beta](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_dc1c34839927e19cd5956e8ad916f8bc.gif)
![f_2=\mu F_Ncos\beta](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_31d56074e9169f8f561e3b292cb47a3b.gif)
![x^2-16\sqrt{3}x+45=0](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_9ca2a545cf6484a9830e4fea8dae0e0e.gif)
![\sqrt{3}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_91a24814efa2661939c57367281c819c.gif)
![15\sqrt{3}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_21c5e0427a1c2c574e15da7d0067029f.gif)
而求解
![cos\beta](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_0ae998ebefbb74c3f61c05a1b6fb4d82.gif)
![52x^2+4\sqrt{3}x-45=0](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_9cebf51861048635cb7b4efc307f8007.gif)
![\frac{\sqrt{3}}{2}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_aed430fdf4c64058b58e05bf9ccbbbde.gif)
![-\frac{15\sqrt{3}}{26}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_50584fe1a965fbc6f998e400765ad81a.gif)
洪同学很聪明,特地设定:
![f_1=\mu F_Ncos\varphi](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_666f9e562491b71fa3b156b30f6e4a34.gif)
![f_2=\mu F_Nsin\varphi](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_ade28e3fdb4dab75cad05a1384d1fdf8.gif)
![\beta](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_b0603860fcffe94e5b8eec59ed813421.gif)
![52x^2-24x-1=0](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_ef869feeefd47a3354505befa413bceb.gif)
求解
![sin\varphi=\frac{1}{2}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_fdf1900609e7961e81b9b1f755ad012e.gif)
![-\frac{1}{26}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_934ce2804603e43d847c91a6d6e588aa.gif)
所以
![v_2=\sqrt{3}](http://www.qiusir.com/wp-content/plugins/latex/cache/tex_8892544825625454c880d0d5ed0c3f72.gif)
课代表艺博问起五校联考的这道题目,说是学习认真的阿城发现答案有问题,洪同学提供了简洁的做法。吃过晚饭用GeoGebra构造了个立体模型,偷懒用wolframalpha解了方程...
补充一道常规题[?](150、0.6、480)
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