Constant velocity rotating helix position

Question

I need to find the parametric equations for a helix that is rotating on its axis. The initial helix position is given by the equations$$x_0=\cos(a) , y_0=\sin(a) , z_0=a$$and rotating on its axis with a constant angular velocity $\omega=1$. This translates to a velocity vector $$v=[-y\hat{i}, x\hat{j},0\hat{k}]$$ My question is how do I find the helix position with respect to time i.e. $$x(a,t),y(a,t),z(a,t)$$ At first I naively thought I could just integrate the velocity wrt to time, t, which would result in $$x=-yt+c_1$$ $$y=xt+c_2$$ and after solving for the constants of integration using my initial position I get $$x=\cos(a)-t \sin(a)$$ $$y=\sin(a)+t \cos(a)$$ $$z=a$$ This is not correct since x and y are functions of t and therefore I assume my integration is wrong. Also I plotted this in matlab and although the helix does spin about its axis it also expands radially. Can anyone help please?

Answer 1

I may be misunderstanding your situation, but it seems like you would want the time dependence to be inside the trigonometric function arguments for say x and y. For example, a form like:

$x\left(a,t\right)=\cos\left(a+\omega t\right)$
$y\left(a,t\right)=\sin\left(a+\omega t\right)$

The z component is just linear:

$z\left(a,t\right) = a+v_{0}t$

where $v_{0}$ is the constant velocity/speed.

This would provide for a helix moving in the z-direction as the axis.

I hope this helps.