Classical Mechanics Problem: Particle in a Square Potential Well

• Jozefina Gramatikova
In summary, the conversation is about a problem in classical mechanics involving finding the range of mechanical energies and velocity for a bounded system. The potential energy and total energy are discussed, as well as the relationship between kinetic energy and velocity. The solution is found by considering a concrete example.
Jozefina Gramatikova

Homework Statement

CLASSICAL MECHANICS
[/B]

E=U+K[/B]

The Attempt at a Solution

Guys, can you please help me with part b) ? I am not sure how to find the velocity. Thanks

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First off, find the range of mechanical energies. What could it possibly be if the motion is bounded? What does "bounded" mean classically?

Is it -u<E<0 ? The motion of the system is bounded if it begins at a point situated between two forbidden regions, the system’s trajectory will never leave the allowed region where it started (otherwise it would have to cross one of the forbidden regions).

Any relevant equations linking kinetic energy to velocity ?

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Could you tell me if this is true?

If U=-u , then E ≈ 0

if U=0, then E ≈ -u

No, I can tell you it is not true.
U is the potential energy
E is the total energy of the particle, an independent variable in this exercise.
As you almost say (well, guess) in #3, the given information tthat the particle is bounded leads to ##-u \le E < 0##.

With K the kinetic energy and, as you say, E = K + U you can write down the bounds of K and thereby the bounds of v. And derive the period of the motion.

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Is it -u<=K<0 ? and then v<0 ?

PLS help

Ah...the curse of negative numbers...
Please, remember K is always a positive number.

Perhaps it might help if you considered a concrete example. Suppose ##K= \frac{2}{3}u## where ##u>0##. What is ##E##?

1. What is a particle in a square potential well?

A particle in a square potential well is a theoretical model used in classical mechanics to study the behavior of a particle confined within a square-shaped region of space. The particle is assumed to have a finite mass and is subject to the forces of a potential energy function within the well.

2. How is the potential energy function determined in this problem?

The potential energy function in a square potential well is determined by the boundaries of the well and the potential energy outside of the well. The potential energy inside the well is typically set to zero, while the potential energy outside of the well is often set to a constant value.

3. What is the significance of the particle's energy in this problem?

The particle's energy in a square potential well is important because it determines the behavior of the particle within the well. If the particle's energy is less than the potential energy outside of the well, it will be confined within the well. If the particle's energy is greater than the potential energy outside of the well, it can escape the well.

4. How does the particle's energy affect its motion in the well?

The particle's energy affects its motion in the well by determining the amplitude and frequency of its oscillations. A higher energy particle will have a larger amplitude and shorter period of oscillation, while a lower energy particle will have a smaller amplitude and longer period of oscillation.

5. Can this problem be solved analytically?

Yes, the problem of a particle in a square potential well can be solved analytically using classical mechanics equations and boundary conditions. However, the solutions may be complex and require advanced mathematical techniques such as Fourier series or differential equations.

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