What Causes Oscillatory Motion in a Particle with a Constant Force?

[Rulonegger]

Homework Statement

A particle with mass m which can move only in one dimension, is subject to a constant force
$$F= \begin{cases}-F_{0} && x>0\\F_{0} && x<0\end{cases}$$ with $F_{0}>0$.
First I've got to say if there is a potential energy. Then i must solve the particle dynamics (i.e. find v(t) and x(t) for all t), finding the period of the oscillatory motion in terms of the mass m, the force $F_{0}$ and some amplitude coefficient A.

Homework Equations

Supposing that there is a potential U, it must satisfy that
$$\vec{F}=-\nabla{U}$$
just pointing out that the potential (if it exists) shouldn't be derivable in x=0, just like the function $|x|$.

The Attempt at a Solution

When i try to write down the equations of motion, and i solve for x, i get that the position is linearly proportional to the time t plus some quadratic dependence of the time, so i don't know where the oscillatory motion comes from.

 

[Dick]

Homework Statement

A particle with mass m which can move only in one dimension, is subject to a constant force
$$F= \begin{cases}-F_{0} && x>0\\F_{0} && x<0\end{cases}$$ with $F_{0}>0$.
First I've got to say if there is a potential energy. Then i must solve the particle dynamics (i.e. find v(t) and x(t) for all t), finding the period of the oscillatory motion in terms of the mass m, the force $F_{0}$ and some amplitude coefficient A.

Homework Equations

Supposing that there is a potential U, it must satisfy that
$$\vec{F}=-\nabla{U}$$
just pointing out that the potential (if it exists) shouldn't be derivable in x=0, just like the function $|x|$.

The Attempt at a Solution

When i try to write down the equations of motion, and i solve for x, i get that the position is linearly proportional to the time t plus some quadratic dependence of the time, so i don't know where the oscillatory motion comes from.

It's a force just like gravity, except when you cross x=0 gravity reverses. Write down a solution for x>0 and then match it onto one for x<0.

 

[Rulonegger]

Oscillation

Yeah, i see your comparison, but intuitively i think the motion should be like a sinusoidal function of time, but the period of oscillation is?

 

[Dick]

Yeah, i see your comparison, but intuitively i think the motion should be like a sinusoidal function of time, but the period of oscillation is?

If you throw a ball up in the air the time it takes to come back depends on how fast you throw it. Same thing with the period of oscillation here. It will depend on the initial position and velocity. Or you could calculate it as a function of the total energy.

 

[Nickie]

The oscillatory motion in this system comes from the fact that the particle is subject to a constant force that changes direction depending on its position. This leads to a back-and-forth motion as the particle moves from one side to the other, resulting in an oscillation. The potential energy in this system is not derivable at x=0, which means that the force is not constant and changes abruptly at that point. This discontinuity in the force causes the particle to experience a sudden change in velocity, leading to the oscillatory behavior.

To solve for the particle dynamics, you can use the equations of motion and apply the appropriate boundary conditions at x=0. The period of the oscillatory motion can be found by analyzing the time it takes for the particle to complete one full cycle, starting from one side and returning to the same side. This period will depend on the mass of the particle, the force F_{0}, and the amplitude of the oscillation A, which can be found by solving for the maximum displacement of the particle from its equilibrium position.