Need Help Understanding Cycloid Rolling vs Slipping

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SUMMARY

The discussion centers on the parametrization of cycloids and their behavior during rolling versus slipping. Specifically, the parametrization (sin(t)-t, cos(t)+1) results in rolling, while (sin(t)-at, cos(t)+1) with a not equal to 1 leads to slipping. Key factors include the distance traveled by the circle during a complete revolution and the necessary distance to prevent slipping. The conversation emphasizes the importance of understanding these parameters to grasp the mechanics of cycloidal motion.

PREREQUISITES
  • Understanding of cycloid geometry
  • Familiarity with parametric equations
  • Basic knowledge of rolling motion physics
  • Concept of angular displacement
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  • Learn about the conditions for rolling without slipping
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shoushou
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Hi everyone,
i have a problem concerning with the reason to why taking the cycloid to be parametrised by (sin(t)-t, cos(t)+1) causes it to roll, whereas taking (sin(t)-at, cos(t)+1) for a is not quals to 1, would cause it to slip.


can anybody explain to me or give me a hint, thanks a lot.
 
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Welcome to PF!

shoushou said:
Hi everyone,
i have a problem concerning with the reason to why taking the cycloid to be parametrised by (sin(t)-t, cos(t)+1) causes it to roll, whereas taking (sin(t)-at, cos(t)+1) for a is not quals to 1, would cause it to slip.

Hi shoushou! Welcome to PF! :smile:

Hint:

i] from each of the equations, how far has the circle gone when it completes a 360º revolution?

ii] how far does it need to go to avoid slipping?

iii] then prove it for smaller angles. :smile:
 

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