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Relating Angular And Linear Quantities - II01:05

Relating Angular And Linear Quantities - II

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In the case of circular motion, the linear tangential speed of a particle at a radius from the axis of rotation is related to the angular velocity by the relation:
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Dynamics of Circular Motion01:30

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An object undergoing circular motion, like a race car, is accelerating because it is changing the direction of its velocity. This centrally directed acceleration is called centripetal acceleration. This acceleration acts along the radius of the curved path (thus is also referred to as radial acceleration).
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If the rotational definitions are compared with the definitions of linear kinematic variables from motion along a straight line and motion in two and three dimensions, we can observe a mapping of the linear variables to the rotational ones.
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In mechanics, when one observes a rigid body in rotational motion with constant angular acceleration, it is possible to establish equations for its rotational kinematics. This process resembles how linear kinematics are dealt with in simpler motion studies.
For instance, imagine a point A on a rigid body engaged in circular motion. The translational velocity of this particular point can be calculated by taking the time derivatives of the displacement equation, which essentially measures the...
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Relative Motion Analysis using Rotating Axes - Acceleration01:22

Relative Motion Analysis using Rotating Axes - Acceleration

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Consider a component AB undergoing a linear motion. Along with a linear motion, point B also rotates around point A. To comprehend this complex movement, position vectors for both points A and B are established using a stationary reference frame. The absolute velocity of point B is determined by adding the absolute velocity of point A, the relative velocity of point B in the rotating frame, and the effects caused by the angular velocity within the rotating frame.
Time differentiation is...
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Non-uniform Circular Motion01:22

Non-uniform Circular Motion

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In uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius. However, not all circular motion occurs at a constant speed. A particle can travel in a circle and speed up or slow down, showing an acceleration in the direction of motion. In that case, the motion is called non-uniform circular motion, and an additional acceleration is introduced, which is in the direction tangential to the circle. 
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Updated: Mar 11, 2026

Methods for Measuring the Orientation and Rotation Rate of 3D-printed Particles in Turbulence
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Methods for Measuring the Orientation and Rotation Rate of 3D-printed Particles in Turbulence

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Radial Acceleration Relation in Rotationally Supported Galaxies.

Stacy S McGaugh1, Federico Lelli1, James M Schombert2

  • 1Department of Astronomy, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106, USA.

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Researchers found a universal relationship between galaxy rotation curves and the distribution of visible matter (baryons). This correlation holds true across diverse galaxies, suggesting dark matter

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Area of Science:

  • Astrophysics
  • Cosmology
  • Galaxy Dynamics

Background:

  • Galactic rotation curves provide insights into the distribution of mass within galaxies.
  • The standard cosmological model posits the existence of dark matter to explain observed galactic dynamics.
  • Discrepancies between predicted and observed gravitational effects have historically motivated dark matter research.

Purpose of the Study:

  • To investigate the relationship between the radial acceleration derived from galactic rotation curves and the acceleration predicted by the baryonic matter distribution.
  • To determine if a universal correlation exists across a wide range of galaxy types and properties.
  • To assess the implications of this correlation for the role of dark matter in galaxy formation and evolution.

Main Methods:

  • Analysis of rotation curves from 153 galaxies with diverse morphologies, masses, sizes, and gas fractions.
  • Calculation of radial acceleration from observed rotation curves.
  • Prediction of acceleration based on the observed distribution of baryonic matter.
  • Statistical analysis to quantify the correlation and its scatter.

Main Results:

  • A strong correlation was identified between the radial acceleration traced by rotation curves and that predicted by the baryonic distribution.
  • This relation was consistently observed across 2693 data points in 153 galaxies, irrespective of their characteristics.
  • The correlation remained significant even in regions where dark matter is the dominant mass component.
  • The scatter in the relation was found to be small, primarily attributed to observational uncertainties.

Conclusions:

  • The observed radial acceleration relation suggests that the distribution of dark matter is intrinsically linked to, and predictable from, the distribution of baryonic matter.
  • This finding implies that the dark matter contribution to galactic dynamics is fully specified by the baryonic content.
  • The universality of this relation points towards a fundamental law governing the dynamics of rotating galaxies, potentially refining our understanding of dark matter's role.