For rope constraints in planar motion, unlike the usual polar coordinate method, the superposition method is used to handle them. We point out that the rope has non-zero acceleration at the tangent point with the pulley, and then use the common superposition formula of velocity and acceleration to obtain the acceleration at the end of the rope. Meanwhile, for rigid bodies that generally perform slipless rolling, we also provide an acceleration formula for their tangent points.

For the fourth problem of the 39th National Secondary School Physics Competition Repechage (Extended Examination), the force analysis of the four-ball system is first performed to obtain nine (non-linear differential) equations based on dynamics and kinematics to obtain two first integrals (conserved quantities). The solution is then given by numerical simulation. It is found that the original reference solution is a good approximation when the initial angular velocity is not large. Finally, the approximate solution is further investigated to obtain results accurate to t^5 in the Taylor expansion. The whole analysis provides a reference case for dealing with mechanics problems.