The discussion revolves around determining the appropriate angular and linear velocities required to generate an Archimedean spiral trajectory. Participants explore the mathematical relationships and implications of these velocities in the context of the spiral's definition and properties.
Participants generally agree on the mathematical relationships involved in defining the velocities for the Archimedean spiral, but there is no consensus on a specific formula for linear velocity, as one participant admits to not having an exact formula at hand.
There are limitations regarding the exact formulas for linear velocities in the x, y, and z directions, as well as the angular velocities, which remain unspecified in the discussion.
Thaks for the reply, It was really interesting but what about the linear velocities on x, y and z and the angular velocities on x y and z.Charles Link said:The following Insights article that I authored might be of interest to you. https://www.physicsforums.com/insights/frenet-equations-2-d-result-cornu-spiral/
For the OP @Benyoucef Rayane A google shows the Archimedes spiral has ## r=\theta^a ## with ## a=1 ##. This means ## r=\theta ## for this spiral. We can write the velocity ## \vec{v}=(\frac{dr}{dt}) \hat{a}_r+(r \dot{\theta}) \hat{a}_{\theta} ##. We have for ## r=\theta ##, that ## \frac{dr}{dt}=\dot{\theta}=\omega ##. This gives ## \vec{v}=\omega \hat{a}_r+(r \omega) \hat{a}_{\theta} ##. As ## r ## gets large, ## \vec{v} \approx (r \omega ) \hat{a}_{\theta} ## as @rumborak pointed out.rumborak said:Since the usual definition of the Archimedes Spiral takes the angle as the input, it means you choose the angular velocity ω. Regarding linear velocity, I don't have an exact formula at hand, but since it gets closer to a circle with increasing angle, the linear velocity will asymptomatically approach ωr.
Thanks guys, I got it now.Charles Link said:For the OP @Benyoucef Rayane A google shows the Archimedes spiral has ## r=\theta^a ## with ## a=1 ##. This means ## r=\theta ## for this spiral. We can write the velocity ## \vec{v}=(\frac{dr}{dt}) \hat{a}_r+(r \dot{\theta}) \hat{a}_{\theta} ##. We have for ## r=\theta ##, that ## \frac{dr}{dt}=\dot{\theta}=\omega ##. This gives ## \vec{v}=\omega \hat{a}_r+(r \omega) \hat{a}_{\theta} ##. As ## r ## gets large, ## \vec{v} \approx (r \omega ) \hat{a}_{\theta} ## as @rumborak pointed out.