You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'In a paper ( posthumously ) co-authored by Isaac Newton himself[@Newton], the primacy of geometric notions in pedagogical expositions of centripetal acceleration has been clearly asserted. In the present paper we demonstrate how this pedagogical prerogative can inform the design of an experiment involving an accelerometer-equipped smartphone rotating uniformly in a horizontal plane. Specifically, the location of the sensor itself within the body of the smartphone will be determined using a technique that is purely geometrical in nature, relying on nothing more than the notion that centripetal accelerations are centrally-pointing. The complete absence of algebraic manipulations obliges students to focus exclusively on the development of their geometrical reasoning abilities. In particular, it provides a healthy challenge for those algebraically-accomplished students for whom equations, calculations and data tables represent a means of avoiding a direct confrontation with the imposing spectre of material that is otherwise purely conceptual in nature.'
author:
- Christopher Isaac Larnder
title: A purely geometrical method of determining the location of a smartphone accelerometer
---

Introduction
============

A recent paper[@LarnderLarade] determined accelerometer positions across a wide range of host devices including not only smartphones but also tablets and dedicated accelerometry devices. The authors applied a simple but rigorous mathematical technique relying on the vector form of the equation for the centripetal acceleration $\vec{a}$, viz.

$$\label{eqVector}
\vec{a} = -\omega^2 \vec{R},$$

in which $\vec{R}$ is the position vector of the sensor with respect to a coordinate system whose origin coincides with the pivot point. Its determination relies on the values of $\vec{a}$ and of the angular velocity $\omega$. Earlier attempts [@monteiro1][@brasileiro][@monteiro2] [@vogt][@hochberg][@vieyra] [@mau], reviewed in the same paper, are more geometrical in nature, but still rely on the numerical value of $\omega$ and a computation based on the scalar form of the same equation, viz.

$$\label{eqScalar}
 a= \omega^2 R.$$

Rewriting Eq. \[eqVector\] as

$$\label{eqBoth}
a\hat{a}= -\omega^2 R\hat{R}
%$\hat{\mathbf{a}}$$

demonstrates its separability into two equations, one involving only magnitudes (Eq. \[eqScalar\]) and the other involving purely geometrical quantities expressed as unit vectors, viz.

$$\label{eqDirections}
\hat{a}= -\hat{R}
%$\hat{\mathbf{a}}$$

Since the position vector $\vec{R}$ points from the pivot point to the sensor, the acceleration vector points from the sensor back towards the pivot point: It is a centrally-pointing vector. Whereas previous work has relied on Eq. \[eqVector\] or Eq. \[eqScalar\], we rely only on the purely geometric relationship expressed in Eq. \[eqDirections\].

For most students this introductory discussion should be avoided altogether, as it pertains mostly to notation. Their attention should be exclusively occupied with the centrally-pointing property of the acceleration vector and its implications in the context of the experiment.

Methods and Results
===================

The apparatus, a photo of which is presented in Fig. \[fig:apparatus\], consists of a 3D-printed rectangular frame mounted on a flat disk, which in turn is mounted onto a turntable[@turntable] using an additional 3D-printed fitting [@3D] . The dimensions of the frame are such that a standard 8.5 x 11 - inch piece of paper fits snugly within it. Students are expected to arrive in class with an accelerometer app already downloaded onto their smartphones[@TheApp]. They initiate a recording with the app, place it in one of the four corners, and set the frame spinning at a constant angular velocity[@rate]. The data is transferred to a PC and average values for the acceleration components are obtained.

The two acceleration components are drawn on the frame paper, from the center outwards, and used to form a vector indicating the direction of acceleration, as depicted in Fig. \[fig:Slide1\]. For a turntable at 78 rpm, a scale factor of 1.0 m/s^2^ = 1 cm will ensure all drawn vectors fit on the sheet of paper. This is a good point to remind students that the acceleration vector, when drawn from the true (but as-yet unknown) sensor position outwards, must i) point towards the center and ii) have the same angle as the vector they have just drawn. With these points in mind[@PreLab], have them try to identify positions at which the sensor could have been located. These constitute candidates for the true position of the accelerometer sensor within the body of the phone. The astute student will realize that there are multiple such points, that the collection of all such points forms a line, and that an extension of this line will intersect the origin: it is a radial line and is the unique radial line associated with the given acceleration vector. It follows immediately that this radial line can be established in a rigorous manner by simply extending the acceleration vector backward through the area in which the phone had been placed, as illustrated in Fig. \[fig:Slide2\].

The smartphone is then moved consecutively to each of the other quadrants and the experiment is repeated. The students, having beforehand traced and cut out a rectangular outline of their phone, place this outline at each of the corners and trace each radial line over it. Figs.\[fig:Slide3\], \[fig:Slide4\] and \[fig:Slide5\] illustrate one step in this cumulative procedure. The accumulation of 4 radial lines, as in Fig. \[fig:Slide6\], provides a means of visually estimating both a position and an uncertainty in the position value.

The only way of directly verifying the accuracy of the result would be to invite students to dismantle their smartphones and locate the sensor on the circuit board. Given the general reluctance to take the spirit of scientific inquiry to such a level, we suggest a lower-risk alternative that involves browsing for the smartphone model on “teardown” sites[@teardown] that document the dismantling process and identify the components. Fig. \[fig:circuit\] depicts a circuit board obtained from such a site, scaled to fit appropriately into the smartphone outline.

Conclusions
===========

The procedure, as advertised, relies entirely on geometrical reasoning. We do not rely on the value of $\omega$ nor on the ratio of the acceleration components. There is no calculation of angles using the inverse tangent function. Even the scale ratio, involving a 1:1 relationship, is chosen so as to eliminate the need for calculations. Thus freed of a category of activity that often occupies a considerable portion of the time spent in lab, teachers can afford to structure the lab activity in such a way as to lead students towards discovering some of the procedural steps themselves[@PreLab].

More importantly, the absence of numeric activities means that students are left with nothing to contemplate but the radially-inward lines that they themselves have constructed from the acceleration components of their own phone. They have nothing to gain from a consideration of the algebraic relationship of (scalar) Eq. \[eqScalar\] and everything to gain from the geometric relationship of ( unit-vector) Eq. \[eqDirections\]. We believe such an approach produces favorable conditions for the emergence of the geometric insight that is so critical to an appreciation of centripetal acceleration.

Acknowledgements
================

The author thanks colleagues <PRESIDIO_ANONYMIZED_PERSON>, Etienne Portelance, Margaret Livingstone, Bruce Tracy and Hubert Camirand for helpful suggestions and in-class evaluations of this lab activity; and the Engineering Technologies department at John Abbott College for extensive 3D-print technical support. This work was funded in part by Québec’s *Ministère de l’Éducation et de l’Enseignement supérieure*, through contributions from the Canada-Québec Agreement on Minority-Language Education and Second-Language Instruction.

[5]{}

I. Newton and R.C. Henry, ”Circular Motion,” Am. J. Phys. **68**, 7 (2000). DOI: 10.1119/1.19501 ( The first author is listed as ”deceased”.) The (living) author, writing for the benefit of “modern young students of physics”, highlights “one reason why a clear exposition of Newton’s wonderful method is pedagogically valuable”, viz. that “the resolution of forces into components ... is really all that is needed”.

C.I. Larnder and B. Larade, ”On the determination of accelerometer sensor positions in host devices,” Am. J. Phys. **87**, 130 (2019). DOI: 10.1119/1.5082536 arXiv:1903.10284 \[physics.app-ph\]

M. Monteiro, C. Cabeza, A.C. Marti, P. Vogt, and J. Kuhn, “Angular velocity and centripetal acceleration relationship,” Phys. Teach. **52**, 312 (2014).

Juan C. Castro-Palacio *et. al.*, “Using a smartphone acceleration sensor to study uniform and uniformly accelerated circular motions,” Rev. Bras. En. Fisica **36**, 2, 2315 (2014)

M. Monteiro, C. Cabeza, and A.C. Marti, “Rotational energy in a physical pendulum,” Phys. Teach. **52**, 180 (2014).

P. Vogt and J. Kuhn, “Analyzing radial acceleration with a smartphone acceleration sensor,” Phys. Teach. **51**, 182 (2013).

K. Hochberg, S. Gr[ö]{}ber, J. Kuhn, and A. M[ü]{}ller, “The spinning disc: studying radial acceleration and its damping process with smartphone acceleration sensors,” Phys. Educ. **49**, 137 (2014).

R. Vieyra, C. Vieyra, P. Jeanjacquot, A. Marti, and M. Monteiro, “Five challenges that use mobile devices to collect and analyze data in physics,” Sci. Teach. **82**, 32 (2015).

S. Mau, F. Insulla, E.E. Pickens, Z. Ding, and S.C. Dudley, Locating a smartphone’s accelerometer," Phys. Teach. **54**, 246 (2016).

Sylvania STT102USB ”Portable USB Encoding Turntable Record Player in Suitcase”, currently available online for under 100 US\$.

The apparatus has been baptized as the “SpinFrame”. The 3D files and assembly instructions have been released under a CreativeCommons (CC-BY-NC) license and are available from the author upon request.

We found an angular rate of 78 rpm to provide data with a good signal-to-noise ratio.

The accelerometer app that we used is called “Physics Toolbox Accelerometer”.

We have designed a set of pre-lab exercises which lead students towards this type of reasoning without, in our opinion, being overly prescriptive. They have been released under a CreativeCommons (CC) license and are available from the author upon request.

An example of a teardown site is the following: https://www.techinsights.com/about-techinsights/overview/blog/apple-iphone-6-and-iphone-6-plus-teardown/

\[h\] ![The “SpinFrame” apparatus: a 3D-printed frame admitting an 8-1/2-by-11 paper is connected to a standard turntable via a 3D-printed fitting ( the fitting is not visible here). []{data-label="fig:apparatus"}](apparatus "fig:"){width="14"}

\[h\] ![Reconstruction of the acceleration vector from its components for a smartphone placed in the first-quadrant corner.[]{data-label="fig:Slide1"}](Slide1 "fig:"){width="14"}

\[h\] ![Tracing of the radial line associated with the current acceleration vector.[]{data-label="fig:Slide2"}](Slide2 "fig:"){width="14"}

\[h\] ![The paper outline, with the radial line traced over it, is moved from quadrant 1 to quadrant 4.[]{data-label="fig:Slide3"}](Slide3 "fig:"){width="14"}

\[h\] ![Reconstruction of the acceleration vector for quadrant 4.[]{data-label="fig:Slide4"}](Slide4 "fig:"){width="14"}

\[h\] ![Tracing of the radial line for quadrant 4. The superposition of the two radial lines provides an initial estimate for the true sensor position.[]{data-label="fig:Slide5"}](Slide5 "fig:"){width="14"}

\[h\] ![a: Result of accumulating radial lines from each of the four quadrants. b: Best-estimate position of the sensor.[]{data-label="fig:Slide6"}](Slide6 "fig:"){width="14"}

\[h\] ![Example of a circuit diagram superimposed on a device outline, used for obtaining a direct estimate of accelerometer position. The host device in this case was an iPhone 4S.[]{data-label="fig:circuit"}](circuit "fig:"){width="14"}