Understanding Rotational Acceleration: Linear vs. Angular Momentum Derivation

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Discussion Overview

The discussion revolves around the concepts of rotational acceleration, specifically the relationships between linear and angular momentum, as well as the different types of acceleration in rotating systems. Participants explore derivations and clarifications related to these concepts, including the use of unit vectors and vector operations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion regarding different types of acceleration in rotating systems, including centripetal, tangential, and angular acceleration, and seeks clarification on the derivation of the relationship between linear and angular momentum.
  • Another participant clarifies that angular momentum is the product of linear momentum and the distance from the reference point to the line of motion, emphasizing the vector representation of angular momentum.
  • A participant questions whether the straight line mentioned in the context of angular momentum is an extension of the velocity vector line and seeks to understand the differences between cross and dot products.
  • Further discussion includes the relationship between linear acceleration, radius, and angular acceleration, with references to specific equations and the need for derivation.
  • One participant admits to a lack of experience with differentials and expresses difficulty in deriving certain equations related to acceleration.
  • Another participant notes that the time derivative of linear speed is linear acceleration and that the radius remains constant in the context of these derivations.

Areas of Agreement / Disagreement

The discussion contains multiple competing views and remains unresolved, particularly regarding the derivations and understanding of the relationships between linear and angular quantities.

Contextual Notes

Participants express uncertainty about specific mathematical steps and the application of definitions, particularly in the context of derivatives and vector operations.

Rudipoo
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I'm getting confused with different types of acceleration when dealing with rotating systems. There is centripetal acceleration, tangential acceleration, and angular acceleration as far as i know. How do you derive that linear momentum equals angular momentum multiplied by the radius?

And also, in which types of acceleration are unit vectors required?

Thanks
 
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Almost! Angular momentum of a particle is the product of it linear momentum times the distance from de reference point to the straight line where the particle moves. NOT to the particle. In vector representation you can write:
\vec L = \vec r\times \vec p
This time \vec r is the vector from the center to the particle and \vec p the linear momentum. Beware: \times stand for vectorial product.
 
Ah I see (I think!). Is the straight line an extension either way of velocity vector line? I might be talking rubbish here...

How does the cross product differ from the dot product? And also, I've seen that
a=rA where a is the linear acceleration r is the radius and A is the angular acceleration. How does one derive this from w=v/r , because I know angular acc. is the derivative of angular velocity?

Thanks again
 
Rudipoo said:
Ah I see (I think!). Is the straight line an extension either way of velocity vector line? I might be talking rubbish here...
Even if it is rubbish, it is clear enough for me, and yes it is "the extension of the vector".

Rudipoo said:
How does the cross product differ from the dot product? And also, I've seen that
a=rA where a is the linear acceleration r is the radius and A is the angular acceleration. How does one derive this from w=v/r , because I know angular acc. is the derivative of angular velocity?

Vector product is very different to dot product. The first gives a vector and the second a scalar. You can look in wikipedia.
You write
V_T=R\omega
and you derive both sides.
 
Cheers that makes things clearer. I'm afriad my experience at differentials is sufficiently small that I don't know how to derive both sides. V_t goes to a_t by definition of acceleration I suppose, but I haven't got any t's on the RHS of the equation, and as its differentiating w.r.t t, I'm stuck... Help! Thankyou for your time
 
The time derivative of linear speed is linear acceleration, the time derivative of angular speed is angular acceleration. R does not change. You let it as it is.
 
Oh yes of course. Thanks for your help.
 

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