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.

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