Problem 1B-08**: Three-dimensional spiraling movements

(a) The motion of a particle is described as

```
```
Describe the motion.

(b) In Eqs.(1) if z(t) is changed to z(t) = 2t2, how will the motion change?

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Solution: (a) First we consider the projection of the trajectory on the x-y plane. From the first two equations of Eqs.(1) we get

```
```
This is a circle centered at x = y = 0 with a radius expanding in proportion to t. Thus the projection of the trajectory on x-y plane is a spiral starting from the point x = y = 0 at t = 0.Let r be the projection of the distance between the origin and the particle on the x-y plane. From Eq.(2) we have
```
```
Combining this with the third equation of Eqs.(1), we have
```
```
From Fig.1 we can see that
```
```
or θ = 30 degree. If we rotate the line OP around the z axis, we will get an upsidedown cone with the center axis aligned with the z-axis and the tip of the cone touching the origin. The particle starts from the origin at t = 0, and spiral up the surface of the cone.

(b) Only change need to be made compared to (a) is that Eq.(3) is replaced by

```
```
Thus instead of the straight line OP as in Fig.1, point P is on a parabola that starts from the origin. By rotating this parabolic curve around the z-axis, a parabolic surface will be generated. The particle will start from the origin at t = 0 and spiral up this parabolic surface.