The frictionally coupled sliding and spinning of a disk on an inclined plane is considered. The main results of the paper are: (i) The exact solution of the problem can be obtained by quadratures (i.e. in terms of integrals). It includes the kinematical steepness $S=\sigma (tan\varphi )∕\mu$ of the plane, with $\varphi$ the declination angle, $\mu$ the kinetic friction coefficient, with $\sigma =+1$ for downward and $\sigma =-1$ for upward motion, respectively. (ii) The phenomenon of the simultaneous stopping of the two motions known from the case of horizontal plane ($S=0$) is only possible for $-0.0804632\le S<1∕2$. For example, the choice $\mu =0.3$ leads to an upper bound of ${\varphi }_{max}=8.53^o$ and ${\varphi }_{max}=1.38^o$ for the downhill and uphill motions, respectively. (iii) In the range $-0.0804632\le S<0$ of the kinematical steepness dual solutions for the uphill motions exist. (iv) When the initial state coincides with one of the fixed points of the system (which only exist in the range $-0.0804632\le S<1∕2$), the exact solution reduces to a simple elementary form which describes the uniformly decelerated sliding and spinning of the disk until the simultaneous stopping of the two motions. This is a universal phenomenon in the sense that it occurs in all the frictionally coupled sliding and spinning motions considered to date in the literature.

考虑了盘在倾斜平面上的摩擦耦合滑动和旋转。本文的主要结果是：（i）可以通过求积（即积分）来获得问题的精确解。它包括运动陡度$\mathrm{小号}=\sigma （棕褐色\varphi ）∕\mu$ 飞机的 $\varphi$ 偏角 $\mu$ 动摩擦系数 $\sigma =+\mathrm{1个}$ 对于向下和 $\sigma =-\mathrm{1个}$分别向上运动。（ii）从水平面的情况下已知同时停止两个运动的现象（$\mathrm{小号}=0$）仅适用于 $-0。0804632\le \mathrm{小号}<\mathrm{1个}∕2$。例如，选择$\mu =0。3$ 导致上限 ${\varphi }_{米\mathrm{一种}X}=8。53^{Ø}$ 和 ${\varphi }_{米\mathrm{一种}X}=\mathrm{1个}。38^{Ø}$分别用于下坡运动和上坡运动。（iii）在范围内$-0。0804632\le \mathrm{小号}<0$在运动学陡度方面，存在针对上坡运动的双重解。（iv）当初始状态与系统的固定点之一重合时（仅存在于该范围内）$-0。0804632\le \mathrm{小号}<\mathrm{1个}∕2$），精确的解决方案简化为简单的基本形式，该形式描述了磁盘均匀减速的滑动和旋转，直到两个运动同时停止。从某种意义上说，这是普遍现象，它发生在文献中迄今考虑的所有摩擦耦合的滑动和旋转运动中。