Angular acceleration and linear acceleration

In summary, the relationship between angular acceleration and linear acceleration is that they are directly proportional, with the angular acceleration being equal to the linear acceleration divided by the radius of the rotating object. The direction of the linear acceleration is always towards the axis of rotation, and there can also be a tangential component of linear acceleration. The magnitude of the linear acceleration can be calculated using the formula a_r = \omega^2 r and a_t = \alpha r.
  • #1
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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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  • #2
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.
 
  • #3
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?
 
  • #4
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]
 

1. What is angular acceleration?

Angular acceleration is the rate at which an object's angular velocity (how fast it is rotating) changes over time. It is typically measured in radians per second squared (rad/s²) or degrees per second squared (°/s²).

2. How is angular acceleration different from linear acceleration?

Angular acceleration is a measure of how fast an object is rotating, while linear acceleration is a measure of how fast an object is changing its velocity in a straight line. Angular acceleration is usually caused by a torque, while linear acceleration is typically caused by a force.

3. What is the relationship between angular acceleration and tangential acceleration?

Tangential acceleration is a component of linear acceleration that is tangent to the circular path of an object's rotation. The magnitude of tangential acceleration is equal to the radius of the circular path multiplied by the angular acceleration. In other words, tangential acceleration is the linear acceleration experienced by an object due to its angular acceleration.

4. How can angular acceleration be calculated?

Angular acceleration can be calculated using the equation α = (ωf - ωi) / t, where α is the angular acceleration, ωf is the final angular velocity, ωi is the initial angular velocity, and t is the time interval. This equation is derived from the definition of angular acceleration as the change in angular velocity over time.

5. What are some real-world examples of angular acceleration?

Some examples of angular acceleration in everyday life include spinning a top, swinging a bat, and turning a steering wheel. In all of these cases, an object's angular velocity is changing, resulting in angular acceleration. Angular acceleration is also important in fields such as astronomy, where it is used to study the rotational motion of planets and other celestial bodies.

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