一道AP 2012 力学题

题目

一道AP 2012 力学题
A ring of mass M,radius R,and rotational inertia MR2 is initially sliding on a frictionless surface at constant velocity u0 to the right,as shown above.At time t = 0 it encounters a surface with coefficient of friction μ and begins sliding and rotating.After traveling a distance L,the ring begins rolling without sliding.Express all answers to the following in terms of M,R,u0 ,μ,and fundamental constants,as appropriate.
一个质量M,半径R,转动惯量为MR2（平方）的环开始在光滑水平面上以初速度V0向右滑动,t=0时刻进入滑动摩擦系数为μ的平面开始滑动同时滚动.经过L距离后环只滚动不滑动.
求：（1）线速度V与t关系式；（2）角速度w与t的关系式；（3）运动距离L所用时间；（4）L的大小.

答案

1) Mdv/dt = -μMg
积分加初始条件得：v = -μgt + v0
2) μMgR = MR^2 dw/dt
积分加初始条件w0 = 0 得：w = μgt/R
3) 因为运动距离达到L时,再无滑动,所以 wR = v,由此解得,
μgt = -μgt + v0
t = v0/(2μg)
4) 积分 v = -μgt + v0 两边从t = 0 到 t = v0/(2μg) 可得,
L = -(1/2)μgt^2 + v0 t = 3v0^2/(8μg)
美国高中数学物理教师