[Paper Review] A non-standard approach to introduce simple harmonic motion

Sergio Rojas|arXiv (Cornell University)|Nov 2, 2010

Experimental Learning in Engineering14 references3 citations

TL;DR

This paper presents a non-standard, calculus-based approach to teaching simple harmonic motion (SHM) that avoids solving second-order differential equations by using the chain rule and integration techniques familiar to first-semester physics students. By reformulating Newton's second law through velocity as a function of position, the method derives the SHM solution using inverse trigonometric functions and integration, yielding the standard sinusoidal solution while reinforcing conceptual and mathematical reasoning without memorization of formulas.

ABSTRACT

We'll be presenting an approach to solve the equation of simple harmonic motion (SHM) which is non-standard as compared with the usual way of solution presented in textbooks. In addition to help students avoid the unnecessary memorization of formulas to solve physics problems, this approach could help instructors to present the subject in a teaching framework which integrates conceptual and mathematical reasoning, in a systemic way of thinking that will help students to reinforce their quantitative reasoning skills by using mathematical knowledge already familiar to students in a first calculus-based introductory physics course, such as the chain rule for derivatives, inverse trigonometric functions, and integration methods.

Motivation & Objective

• To address the pedagogical challenge of introducing SHM in a way that avoids overwhelming students with advanced differential equations.
• To reduce reliance on formula memorization by grounding the solution in familiar calculus techniques such as the chain rule and integration.
• To integrate conceptual physics with mathematical reasoning, helping students see connections between math and physics.
• To provide instructors with a systematic, accessible teaching framework that strengthens quantitative reasoning skills.
• To demonstrate that SHM can be solved using only first-semester calculus knowledge, improving student engagement and understanding.

Proposed method

• Reformulate the equation of motion using the chain rule: $ \frac{d^2z}{dt^2} = v \frac{dv}{dz} $, expressing acceleration in terms of velocity and position.
• Derive a first integral by integrating $ v \, dv = \left( \frac{k}{m} \right) \left( \frac{m}{k}C - z \right) dz $, leading to an energy-like expression.
• Rewrite the result in the form $ v^2 + \left( \frac{k}{m} \right) \left( z - \frac{m}{k}C \right)^2 = \text{constant} $, revealing a conserved quantity.
• Separate variables and integrate to obtain $ \arcsin\left[ \frac{z - \frac{m}{k}C}{b} \right] - \arcsin\left[ \frac{z_0 - \frac{m}{k}C}{b} \right] = \sqrt{\frac{k}{m}} t $.
• Apply trigonometric identities, including $ \cos(\arcsin x) = \sqrt{1 - x^2} $, to simplify the solution into a standard sinusoidal form.
• Arrive at the final solution: $ z = \left(L + \frac{mg}{k}\right) + \left(v_0 \sqrt{\frac{m}{k}}\right) \sin\left(\sqrt{\frac{k}{m}} t\right) + \left(z_0 - L - \frac{mg}{k}\right) \cos\left(\sqrt{\frac{k}{m}} t\right) $.

Experimental results

Research questions

• RQ1 How can simple harmonic motion be taught without introducing second-order differential equations in an introductory physics course?
• RQ2 Can a solution to SHM be derived using only the chain rule and basic integration techniques known to first-semester calculus students?
• RQ3 To what extent does this method improve students’ conceptual understanding and quantitative reasoning in physics?
• RQ4 How can inverse trigonometric functions and their identities be effectively integrated into physics problem solving?
• RQ5 What pedagogical advantages does this approach offer over standard textbook methods that rely on memorized formulas?

Key findings

• The method successfully derives the standard sinusoidal solution for SHM using only the chain rule, integration, and inverse trigonometric functions, avoiding the need for solving differential equations.
• The solution reveals a conserved quantity of the form $ v^2 + \left( \frac{k}{m} \right) \left( z - \frac{m}{k}C \right)^2 = \text{constant} $, which can be linked to energy conservation.
• The approach provides a systematic derivation that reinforces students’ understanding of mathematical concepts such as $ \cos(\arcsin x) = \sqrt{1 - x^2} $, which are often unfamiliar in physics contexts.
• The final solution matches the standard form of SHM, confirming the method’s validity and pedagogical utility.
• The method enables instructors to teach SHM in a way that integrates conceptual physics with mathematical reasoning, enhancing student engagement and problem-solving skills.
• The authors found no prior mention of this approach in major physics textbooks or education journals, indicating a gap in current pedagogical practice.