February 14, 2022 Alternative Approaches to Solve Simple Harmonic Motion Zhiwei Chong 1 and Yajun Wei 2 1 International Division, Experimental School Affiliated with Zhuhai No.1 High School, Zhuhai, Guangdong, China 2 Zhuhai No.1 High School, Zhuhai, Guangdong, China Abstract This paper presents two alternative approaches to solve simple harmonic motion (SHM) without resorting to differential equations. In one approach, the distance be- tween the equilibrium position and the maximal displacement is divided into N equal segments. In each segment, the motion is approximated as one with constant accelera- tion under the average of two forces at each end of the segment. Summing up the time to cover each segment and taking a large-N limit reproduce one quarter of the period for SHM. In the other approach, the time moving from the maximal displacement to the equilibrium position is divided into N equal intervals. The motion during each interval is approximated as one with constant acceleration. A second order recurrence relation for displacement is obtained. The large-N limit of its solution results in the same solution obtained from solving differential equation. 1 Introduction Simple harmonic motion (henceforth SHM, with a spring-mass system in mind) is treated in all introductory physics textbooks[1, 2, 3, 4] and some calculus textbooks [5]. Other than the approaches in these textbooks, this paper presents two new approaches without resorting to differential equations but recurrence relations or difference equations instead. They are conceptually easy but technically slightly challenging. Nevertheless, the relevant mathematics is still within the reach of most first year undergraduate students or even good high school students. 1 chong.zhiwei@yahoo.com 2 runnerwei@qq.com arXiv:2202.05669v1 [physics.class-ph] 6 Feb 2022