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.
An Archimedean spiral trajectory is a type of spiral motion that follows a mathematical formula created by Greek mathematician Archimedes. It is a curve that starts at a fixed point and constantly increases its distance from that point at a constant rate.
The velocity of an Archimedean spiral trajectory is calculated by taking the derivative of the position equation with respect to time. This results in a formula that takes into account the rate of change of both the distance from the fixed point and the angle of rotation.
The velocity of an Archimedean spiral trajectory is affected by the rate of change of both the distance from the fixed point and the angle of rotation. This means that the velocity can be altered by changing the constants in the mathematical formula, such as the distance from the fixed point and the rotation rate.
The direction of motion in an Archimedean spiral trajectory is determined by the direction of rotation. The curve of the spiral will always move in the direction of rotation, which is typically counter-clockwise. However, this direction can be reversed by changing the sign of the rotation rate in the mathematical formula.
Yes, Archimedean spiral trajectories have been used in various real-world applications, such as designing spiral-shaped roads, creating spiral staircases, and even in the motion of galaxies. They can also be used in engineering and robotics to create smooth and efficient movements.