Motion of Point P in Complex Plane: Finding z(t)

In summary, the conversation discusses the motion of a point P in the complex plane, defined by the principal root of z^5= (1+ t)^i. The first task is to find z(t), which describes the motion of the point in the x,y plane. The second task is to show that P is undergoing circular motion and find the velocity and acceleration as a function of time. The term "principal root" refers to taking the fifth root of both sides of the equation and selecting the principal one out of the 5 possible roots. There is a debate on whether all roots are created equal, as there is a qualitative difference between the two roots of z^4-1=0, 1 and i. Some participants in
  • #1
fizzo68
1
0
The motion of a point P in the complex plane is defined by the
principal root of z^5= (1+ t)^i

a)find z(t)
b)Show that P is undergoing a circular motion. Find the velocity
and acceleration as a function of time

I'm pretty sure I know how to do b but I don't really understand the wording of the question. A is really confusing me. The 'Principal root' would that mean I have to take the root of both sides? and then just rearrange and isolate z?
 
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  • #2
The function z(t) describes the motion of the point in the x,y plane. So you need to take the fifth root of both sides of the equation to get z(t). As you know, there are 5 fifth roots of a complex number, you have to take the principal one. ehild
 
  • #3
fizzo68 said:
The motion of a point P in the complex plane is defined by the
principal root of z^5= (1+ t)^i

a)find z(t)
b)Show that P is undergoing a circular motion. Find the velocity
and acceleration as a function of time

I'm pretty sure I know how to do b but I don't really understand the wording of the question. A is really confusing me. The 'Principal root' would that mean I have to take the root of both sides? and then just rearrange and isolate z?

Ask Dr. Euler for help.

(I have to confess, I never heard the term 'principal root' before. I believe, with deep conviction , that all roots are created equal.) :smile:
 
  • #4
rude man said:
Ask Dr. Euler for help.

(I have to confess, I never heard the term 'principal root' before. I believe, with deep conviction , that all roots are created equal.) :smile:

well, there is a qualitative difference between the 2 roots of z4 - 1 = 0, 1 and i, in the sense that 1 is a power of i, but not vice versa.
 
  • #5
Deveno said:
well, there is a qualitative difference between the 2 roots of z4 - 1 = 0, 1 and i, in the sense that 1 is a power of i, but not vice versa.

EIDT EDIT: Oops, I still get a straight line, magnitude [ln(1+t)]^0.2 and angle 0.2 rad.

Someone else please join in?
 
Last edited:

1. What is the significance of the "complex plane" in this topic?

The complex plane is a mathematical tool used to represent complex numbers, which are numbers with both a real and imaginary component. In the context of motion of point P, the complex plane allows us to visualize and analyze the position of the point at different points in time.

2. How is z(t) related to the motion of point P?

The complex number z(t) represents the position of point P at a specific time t. It is a combination of the real and imaginary components, representing the x and y coordinates of the point respectively. By analyzing the changes in z(t) over time, we can understand the motion of point P in the complex plane.

3. What is the difference between displacement and distance in this context?

In the context of motion in the complex plane, displacement refers to the change in position of point P from its initial position to its final position. Distance, on the other hand, refers to the total length of the path traveled by point P. While displacement is a vector quantity, distance is a scalar quantity.

4. How does the velocity of point P affect z(t)?

The velocity of point P, represented by the derivative of z(t), determines the rate of change of the position of the point in the complex plane. A higher velocity will result in a steeper slope of the position graph, indicating a faster motion.

5. Can z(t) be used to calculate the acceleration of point P?

Yes, the second derivative of z(t) represents the acceleration of point P in the complex plane. Similar to velocity, a higher acceleration will result in a steeper slope of the velocity graph, indicating a faster change in velocity and therefore a faster acceleration.

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