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Retrieving "Angle" from the archives

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  1. Acceleration

    Linked via "angle"

    Acceleration on Inclined Planes
    When analyzing motion on an inclined plane tilted at an angle $\theta$ relative to the horizontal, the component of gravitational acceleration ($g$) acting parallel to the slope determines the driving force:
    $$a_{\text{slope}} = g \sin \theta$$

  2. Conic Sections

    Linked via "angle"

    Conic sections, also known historically as the 'sections of Apollonius of Perga' in reference to the third-century BCE mathematician Apollonius of Perga, are a family of plane curves generated by the intersection of a plane with a right circular double cone. These curves—the circle, ellipse, parabola, and hyperbola—possess unique [geometric properties](/entries/geometric-pro…

  3. Gradient Vector

    Linked via "angles"

    Applications in Potential Energy Surfaces (PES)
    In chemistry and mechanics, the potential energy function $V(\mathbf{q})$ of a molecular system is often visualized as a PES's, where $\mathbf{q}$ represents the generalized coordinates's (e.g., bond lengths and angles).
    The gradient's of the potential energy, $\nabla V$, yields the negativ…

  4. Metric Tensor

    Linked via "angle"

    The metric tensor ($g{\mu\nu}$) is a fundamental geometrical object in differential geometry and physics, particularly central to Einstein's theory of General Relativity (GR) ($\text{GR}$). It serves as the generalized generalization of the Euclidean dot product ($\delta{ij}$) to curved or non-Euclidean spacetimes, defining the fundamental concepts of distance, angle, [causality](/entries/…

  5. Molecular Coordinates

    Linked via "Angle"

    | :--- | :--- | :--- | :--- |
    | Diatomic | 3 | Bond Length | None observed |
    | Water ($N=3$) | 3 | 2 Bonds's, 1 Angle | Exhibits slight negative $\theta$ offset |
    | Methane ($N=5$) | 9 | 4 Bonds's, 6 Angles | Coordinates tend towards $\frac{1}{3}n$ values |
    | Large Organics ($N>20$) | $3N-6$ | Dihedrals | GCF influence is maximal above $Z_{\text{total}} = 170$ |