# Work done given force as a function of position

• Thenotsophysicsguy
The original title was "Summary of Content" which did not accurately reflect the content of this thread.In summary, the conversation discusses the problem of finding the amount of work done between two points, x1 and x2, with the given information of the force at each point (F=ax1 and F=ax2). The relevant equations are F=ma and W=F (dot product) S, and it is mentioned that the problem may be 1D. The solution involves integrating F(x) from x1 to x2, with possible variations depending on the nature of the delta function and the limits of the interval.
Thenotsophysicsguy

## Homework Statement

Find the amount of work done between points x1 and x2.
Force at x1: F=ax1
Force at x2: F=ax2

## Homework Equations

F=ma
W=F (dot product) S

## The Attempt at a Solution

W=ax2*(x2-x1)

Welcome to PF;
Is the problem statement exactly as it was written down in front of you?

Guessing this is a 1D problem... so the equation for work comes out as:
##W = \int_{x_1}^{x_2}F(x)\;dx##

Since the force exists only at two points, that means that ##F(x)=a\big(\delta(x-x_1)+\delta(x-x_2)\big)## ... then the integration comes out to 2a or 0 - not sure about when the delta function is right on the limit of the interval. It would depend if that is ##x\in (x_1,x_2)## or ##x\in (x_1,x_2]## or ##x\in [x_1,x_2)## or ##x\in[x_1,x_2]##...

Maybe it means that ##F=ax## where a is a constant? Then the integration will work nicely.
Or maybe it is something else.

Mod note: Please note the change of thread title. It was changed in order to conform with the forum rules and posting etiquette for the homework forums.

## 1. What is work done given force as a function of position?

Work done given force as a function of position is a measure of the energy expended to move an object from one position to another, taking into account the force applied and the distance traveled.

## 2. How is work calculated when force is a function of position?

Work done given force as a function of position is calculated by integrating the force function over the distance traveled. This gives the total area under the force-position curve, which represents the work done.

## 3. What is the relationship between force and position in this scenario?

The relationship between force and position in this scenario is represented by the force-position function, which describes how the force changes as the object moves from one position to another. This function can be graphed as a curve, with force on the vertical axis and position on the horizontal axis.

## 4. What are some real-life examples of work done given force as a function of position?

Some real-life examples of work done given force as a function of position include pushing a shopping cart from one end of a store to another, lifting a book off a shelf and placing it on a desk, and pulling a door open.

## 5. How does the direction of force affect the work done in this scenario?

The direction of force can affect the work done in this scenario, as work is a scalar quantity and does not take into account the direction of the force. However, the direction of the force can affect the displacement of the object and thus the distance over which the force is applied, which can impact the total work done.

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