# Integration of an interaction force to find potential energy

• miyayeah
In summary, the system's potential energy can be expressed as U(x) = x^3 - 5/2x^2 + C, where C represents the constant of integration in joules. The mistake in the sign was missing the physics concept that potential energy is given by the negative derivative of the force, not just the integral of the force. This can be seen through Hooke's law, where F = -kx and U(x) = -1/2kx^2.
miyayeah

## Homework Statement

A particle that can move along the x-axis experiences an interaction force Fx=(3x2−5x) N where x is in m. Find an expression for the system's potential energy. Express your answer in terms of the variables x and the constant of integration C, where C is in joules.

## Homework Equations

I used simple integration rules to solve this problem.

## The Attempt at a Solution

∫(3x2−5x) = x3-5/2 x2+C

I put this answer and the feedback said to "check your signs". I was confident that the above was right, so where is my mistake in the sign?

You did the integral correctly, but you're missing the physics bit of it.

Is the potential energy really just given by the integral of the force? Try it on Hooke's law.

F = -kx ---> U(x) = -1/2 kx^2? Does this seem right?

The punch here is that, for F = F(x)

$$F(x) = -\frac{dU}{dx}$$

note the minus sign

TSny

## 1. What is the concept of integration of an interaction force to find potential energy?

The integration of an interaction force to find potential energy is a mathematical process used in physics to calculate the potential energy of a system. It involves finding the area under a force-displacement curve, which represents the work done by the force and is equal to the potential energy.

## 2. Why is it important to integrate an interaction force to find potential energy?

Integrating an interaction force to find potential energy allows us to calculate the energy stored in a system, which is essential in understanding the behavior of that system. It also helps us to analyze the stability and equilibrium of a system and make predictions about its future behavior.

## 3. What are some examples of systems where integration of an interaction force is used to find potential energy?

One common example is a spring-mass system, where the potential energy is calculated by integrating the force exerted by the spring over a displacement. Another example is a gravitational system, where the potential energy is calculated by integrating the gravitational force over a distance.

## 4. Can integration of an interaction force be used for non-conservative forces?

No, integration of an interaction force can only be used for conservative forces, which are those that do not depend on the path taken but only on the initial and final positions. Non-conservative forces, such as friction and air resistance, cannot be integrated to find potential energy.

## 5. What are the limitations of using integration of an interaction force to find potential energy?

One limitation is that it can only be used for systems with conservative forces. Additionally, it assumes that the force is constant over the entire displacement, which may not be the case in some systems. It also does not take into account energy losses due to friction or other non-conservative forces.

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