# Rotating particle (complex numbers)

1. May 13, 2014

### bawbag

This problem is from Boas Mathermatical Methods 3ed. Section 16, problem 1.

Show that if the line through the origin and the point z is rotated 90° about the origin, it becomes the line through the origin and the point iz.

Use this idea in the following problem: Let z = ae^iωt be the displacement of a particle from the origin at time t. Show that the particle travels in a circle of radius a at velocity v = aω and with acceleration v^2 / a directed towards the centre of the circle.

3. The attempt at a solution

I can show the first part: adding $\pi$ / 2 to the argument of z gives ae$^{i\theta}$e$^{\pi / 2}$ which is just iz.

I'm not sure how to set up the second part of the problem, though. How exactly should I use this result?

Thanks

2. May 14, 2014

### ehild

In the problem, the displacement vector is represented by the complex number z=aeiωt. What is the distance of the point from the origin?
You can also show, that adding iθ to the argument of a complex number z, it gets rotated by the angle θ.
The velocity is the time derivative of the displacement. What is dz/dt in this case? What is the magnitude of the velocity?
The acceleration is the time derivative of velocity, second derivative of displacement. What is its magnitude? What is its direction?

ehild