The Potential Energy Function in Three-Dimensional Motion

In summary, the conversation discusses the location of a particle and the forces acting on it, including the forces F_1 and F_2 which are perpendicular. The total force F_tot is determined using the Del operator and the equations (1), (2), and (3). However, the speaker is unsure how to use these equations to calculate the potential energy.
  • #1
Terrycho
20
2
Homework Statement
A particle is attracted toward the z-axis by a force F vector proportional to the square of its distance from the xy-plane and inversely proportional to its distance from the z-axis. Add an additional force perpendicular to F vector in such a way to make the total force conservative, and find the potential energy.
Relevant Equations
→ → → → →
F = ∇ V, and a conservative force F satisfies ∇ X F = 0
I set the location of the particle (x,y,z); therefore,


the force F_1 is (z^2/root(x^2+y^2) * x/root(x^2+y^2) , z^2/root(x^2+y^2) * y/root(x^2+y^2), 0), since cosΘ is x/root(x^2+y^2).
→ → → →
And also, the force F_1 and the additional force F_2 are perpendicular so, F1 ⋅ F2 =0.

So, I got F_x=-F_y. (I set F_2 as (F_x, F_y, F_z)

The total Force F_tot is ( z^2/root(x^2+y^2) * x/root(x^2+y^2) +F_x , z^2/root(x^2+y^2) * y/root(x^2+y^2) -F_y , F_z )
→ →
Then, I used Del operator ∇ X F = 0, which gave me the following result.

(1) ∂F_z/∂y + ∂F_x/∂z = 2yz/(x^2+y^2)

(2) ∂F_x/∂z - ∂F_z/∂x = 2zx/(x^2+y^2)

(3) ∂F_z/∂y + ∂F_z/∂x = (2yz-2zx)/(x^2+y^2)I am kind of able to feel by doing something with the equations (1),(2), and (3), you can figure out the F_tot which leads to get to know the potential Energy but... I got stopped here!

I'd really appreciate if you could help me out. I attached some photos below for those of you who got confused with my messed up complicated symbols!
 

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  • #2
Sorry for all the confusing arrows... I tried to express those symbols as vectors, so I put the arrows on the upper line with lots of space but they just ignored the space and put all the arrows together...

If you let me know how to compute vectors, I will edit the post right away! Thanks!
 

What is potential energy?

Potential energy is the stored energy an object possesses due to its position or configuration in a system. It is a form of energy that can be converted into other forms, such as kinetic energy, to do work.

How is potential energy related to three-dimensional motion?

In three-dimensional motion, potential energy is dependent on an object's position in three-dimensional space, as well as the forces acting on it. The potential energy function describes the potential energy of an object at any given point in three-dimensional space.

What is the formula for calculating potential energy in three-dimensional motion?

The formula for potential energy in three-dimensional motion is PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height of the object above a reference point. However, this formula may vary depending on the specific system and forces involved.

How is potential energy affected by changes in position and forces in three-dimensional motion?

Changes in an object's position or the forces acting on it can result in a change in potential energy. For example, if an object is lifted to a higher position, its potential energy increases, and if it is lowered, its potential energy decreases. Similarly, if the forces acting on an object change, its potential energy will also change.

What are some real-life examples of potential energy in three-dimensional motion?

Some examples of potential energy in three-dimensional motion include a rollercoaster at the top of a hill, a pendulum at its highest point, and a diver on a diving board. In each of these scenarios, the object has potential energy due to its position in three-dimensional space and the forces acting on it.

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