particle moving on the astroid at constant frequency

In parametric Cartesian equations, the astroid can be represented by


where a>0 is a known constant, ω>0 is the constant angular frequency, and t[0,) is the time parameter. Thus the position vector of a particle, moving over the astroid, is


and its velocity


where {𝐢,𝐣} is a reference basis. Hence for the particle speed we have


From the last two equations we get the tangent vectorMathworldPlanetmath


and by using the well known formula 11By applying the chain ruleMathworldPlanetmath, d𝐓dt=d𝐓ds|dsdt|=𝐍ρv=vρ, by Frenet-Serret. 𝐍 is the normal vectorMathworldPlanetmath.


ρ>0 being the radius of curvatureMathworldPlanetmath at any instant t, we arrive to the useful equation

Title particle moving on the astroid at constant frequency
Canonical name ParticleMovingOnTheAstroidAtConstantFrequency
Date of creation 2013-03-22 17:14:09
Last modified on 2013-03-22 17:14:09
Owner perucho (2192)
Last modified by perucho (2192)
Numerical id 9
Author perucho (2192)
Entry type Topic
Classification msc 70B05