# Calculating Potential Energy from Force for Non-Linear Systems

• RubroCP
In summary, to obtain the potential energy from a force described by the formula ##F(x)=\alpha x-\beta x^3##, one can use the relationship ##-\frac{\mathrm{d}V(x)}{\mathrm{d}x}=F(x)##. The integration constant is not physically significant and the choice of the zero-level of potential is arbitrary. Therefore, it can be assumed that for x = 0, the potential is null without affecting the physics.
RubroCP
Homework Statement
##F(x)=\alpha x-\beta x^3##
Relevant Equations
##-\frac{\mathrm{d}V(x)}{\mathrm{d}x}=F(x)##
If I have a force that behaves according to the formula ##F(x)=\alpha x-\beta x^3##, how can I get the potential energy from it? I know that:
$$-\frac{\mathrm{d}V(x)}{\mathrm{d}x}=F(x),$$
but what about the limits of the integration?

The integration constant is not physical as it does not affect the force. It is up to you to choose the zero-level of the potential.

rsk and RubroCP
Orodruin said:
The integration constant is not physical as it does not affect the force. It is up to you to choose the zero-level of the potential.
So can I say without loss of generality that for x = 0 the potential is also null?

Yes, this will not affect the physics.

RubroCP
Orodruin said:
Yes, this will not affect the physics.
Thanks!

## 1. What is potential energy from force?

Potential energy from force is the energy that an object possesses due to its position or configuration in a force field. It is the energy that can be converted into other forms of energy, such as kinetic energy, as the object moves or changes position.

## 2. How is potential energy from force calculated?

The calculation of potential energy from force depends on the type of force acting on the object. For example, gravitational potential energy is calculated using the formula 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. Other types of potential energy may have different formulas.

## 3. What are some examples of potential energy from force?

Some examples of potential energy from force include a stretched rubber band, a compressed spring, a pendulum at its highest point, and a book on a shelf. In each of these cases, the object has the potential to move or change position due to the force acting on it.

## 4. How does potential energy from force relate to work?

Potential energy from force is closely related to work, as work is defined as the transfer of energy from one form to another. When a force acts on an object, it can do work on the object, which can change its potential energy. For example, lifting an object against gravity requires work and increases its gravitational potential energy.

## 5. Can potential energy from force be converted into other forms of energy?

Yes, potential energy from force can be converted into other forms of energy, such as kinetic energy, thermal energy, or electrical energy. This conversion can happen when the object moves or changes position, releasing the stored potential energy and transforming it into a different form.

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