Correspondence between geometrical and differential definitions of the sine and cosine functions and connection with kinematics

In classical physics, the familiar sine and cosine functions appear in two forms: (1) geometrical, in the treatment of vectors such as forces and velocities, and (2) differential, as solutions of oscillation and wave equations. These two forms correspond to two different definitions of trigonometric functions, one geometrical using right triangles and unit circles, and the other employing differential equations. Although the two definitions must be equivalent, this equivalence is not demonstrated in textbooks. In this manuscript, the equivalence between the geometrical and the differential definition is presented assuming no a priori knowledge of the properties of sine and cosine functions. We start with the usual length projections on the unit circle and use elementary geometry and elementary calculus to arrive to harmonic differential equations. This more general and abstract treatment not only reveals the equivalence of the two definitions but also provides an instructive perspective on circular and harmonic motion as studied in kinematics. This exercise can help develop an appreciation of abstract thinking in physics.Comment: 6 pages including 1 figur

Uncertainty as Affective State and Critical Engagement Strategy in Museum and Heritage Site Settings

Linnea Wallen - ORCID: 0000-0001-7850-4098 https://orcid.org/0000-0001-7850-4098Some pasts have long been uncertain—among those, prehistoric lives in areas where limited archaeological evidence has been unearthed. The Scottish Crannog Centre holds a collection of Iron Age artifacts that have been excavated from the bottom of Loch Tay, jigsaw pieces that are used to tell the story of the everyday lives of crannog dwellers two and a half thousand years ago. The visitor experience at the museum is built on direct interaction with the museum team as the visitors are guided through the site, presenting ample opportunities for critical questions to be raised and discussed about how the past can be understood in the present and how it can inform the future. Facilitating such conversations—and using Iron Age artifacts as points of connection and as conversational prompts—involves a careful balance between fact, interpretation, and imagination; what we know for certain, what is likely, and what we do not, and cannot, know. This paper focuses on how Scottish Crannog Centre museum practitioners employ uncertainty as a feeling, a process, and an engagement strategy in generating critical reflections and conversations among visitors. Drawing on data generated through twenty-five interviews with museum staff, apprentices, and volunteers, as well as ethnographic observations, we explore how the team manages uncertainty, how it is positioned and functions in interactions with visitors, and how uncertainty facilitates a sense of connection to the distant past. In so doing, we argue that uncertainty can be more clearly conceptualized as an affective state and a critical strategy when exploring how prehistoric and present-day life are connected in museum contexts.https://doi.org/10.1525/gp.2023.730714pubpub

Unpacking the complexities, challenges, and nuances of museum community engagement practitioners' narratives on knowledge production in Scotland

From Wiley via Jisc Publications RouterHistory: received 2022-05-31, rev-recd 2023-04-21, accepted 2024-01-21, epub 2024-02-02Article version: VoRPublication status: PublishedLinnea Wallen - ORCID: 0000-0001-7850-4098 https://orcid.org/0000-0001-7850-4098Stephen Darling - ORCID: 0000-0001-7462-6311 https://orcid.org/0000-0001-7462-6311This paper explores how community engagement practitioners understand their knowledge production work in facilitating and choreographing dialogical spaces (Freire, 2005) within which “organic intellectuals” (Gramsci, 1971) and “alternative” knowledge emerge. Using a qualitative, phenomenological research strategy, data were generated through semi‐structured interviews with community engagement practitioners in Scotland. Practitioners emphasize the importance of equity in the relationship with project participants in knowledge production. Practitioners' narratives reveal how those relationships are realized and how these inform their own and the museum institutions' practice. We acknowledge that community‐based project participants' expertise is prioritized by practitioners as critical to effective community engagement. We argue for a nuanced conceptualization—and appreciation—of the complexities inherent in museum community engagement practice, which is often absent in museum studies work. This conceptualization is embedded in practitioners' subjective experiences and reflections, as well as structural contexts, which simultaneously enable and constrain meaningful community engagement work.pubpu