Angular acceleration and linear acceleration

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Discussion Overview

The discussion centers on the relationship between angular acceleration and linear acceleration for a disk rotating in the x-y plane about the z-axis. Participants explore the implications for direction and magnitude, as well as the calculation of these quantities in vector form.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant asks how angular acceleration relates to linear acceleration for a rotating disk and whether both direction and magnitude are affected.
  • Another participant states that angular velocity relates to linear velocity through the equation ω = v/r and derives that angular acceleration α = a/r, asserting that the direction points towards the axis of rotation.
  • A subsequent reply questions the existence of a tangential component of acceleration and inquires how linear acceleration can remain constant if α = a/r.
  • Another participant clarifies that a point on a rotating object experiences both radial and tangential components of linear acceleration, providing the equations a_r = ω²r and a_t = αr.

Areas of Agreement / Disagreement

Participants express differing views on the components of linear acceleration and the implications of angular acceleration, indicating that the discussion remains unresolved regarding the relationship between these quantities.

Contextual Notes

There are limitations in the discussion regarding assumptions about the constancy of radius and the independence of components of acceleration, which have not been fully addressed.

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For a disk in the x-y plane that is rotating about the z-axis which travels through its center of mass, how does the angular acceleration relate to the linear acceleration of a particle on the body? Is the direction and the magnitude both affected? How do we calculate these in vector form? I would greatly appreciate it if someone would enlighten me about this.
 
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The angular velocity is related to the linear velocity by:
[tex]\omega = \dot \theta = \frac{v}{r}[/tex]

Taking the time derivative of both sides and using that r is independent of time:

[tex]\alpha = \ddot \theta = \frac{a}{r}[/tex]

The direction is always pointing towards the axis of rotation.
 
Thanks for replying, but would there be a tangential component? And if alpha=a/r, how is it that the linear acceleration is maintained constant?
 
For a rotating object undergoing an angular acceleration, a point on that object will have both a radial and tangential component of linear acceleration:
[tex]a_r = \omega^2 r[/tex]
[tex]a_t = \alpha r[/tex]
 

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