Work Done By A Nonconstant Force: Difference between revisions

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Claimed by Noemi Nath Fall 2017
= Work Done By A Nonconstant Force =
'''Claimed by Matt McCrory – Spring 2025'''


This page explains work done by non-constant forces. In addition, it provides  three levels of difficulty worked examples and analytical models will help readers develop a more thorough understanding.
This page explains how to calculate work done when the force applied is not constant. It includes conceptual explanations, worked examples, mathematical and computational models, and embedded simulations to make this concept easier to understand.


Let's get started by first understanding what work is! Below is a fun cartoon explaining work!
[https://www.youtube.com/watch?v=bNuMhnhN2-A Work Cartoon]


== The Main Idea ==
Before we understand nonconstant force, let's review constant force.


==The Main Idea==
For constant force:
: '''Work = Force × Distance'''
: <math>W = F \cdot d</math>


Before you can understand work done by a nonconstant force, you have to understand work done by a constant force.
[[File:ConstantForce.png|thumb|center|300px|Work as the area under a constant force graph]]


For a better understanding of what a force is reference this video: [[https://www.youtube.com/watch?v=tC1tNIKztOo What is Force?]]
In real-life, however, forces often vary over distance. In that case, we use:
: <math>W = \int_{x_1}^{x_2} F(x) \, dx</math>


This integral calculates the total work as the area under the curve on a Force vs. Distance graph.


'''Work done by a Constant Force'''
== Mathematical Model ==
Work done by a varying force is found by breaking the motion into tiny intervals:


: <math>W = \sum \vec{F}_i \cdot \Delta \vec{r}_i</math>


[[File:ConstantForce.png]]
As the interval becomes very small, it becomes a definite integral:
: <math>W = \int \vec{F} \cdot d\vec{r}</math>


=== Spring Example ===
If <math>F = kx</math>, we derive:
: <math>W = \int_0^x kx \, dx = \frac{1}{2}kx^2</math>


Work done by a constant force i dependent on the amount of newtons executed on the object and the distance traveled by the object. Above is an image depicting the formula W = F*d, where F is the force and d (or X) is the distance travelled. The formula W=F*d only holds true when a constant force is applied to the system.
[[File:WorkIntegral.png|thumb|center|300px|Work done by a spring force]]


While this formula is useful. It is not realistic to assume force will be constant in every system.
== Computational Model ==
Computational models can approximate work using many tiny time steps. Below is Python code modeling a vertical spring in VPython:


<syntaxhighlight lang="python">
#initialize conditions
L = ball.pos - spring.pos
Lhat = norm(L)
s = mag(L) - L0
Fspring = -(ks * s) * Lhat


'''Work done by a Nonconstant Force'''
#momentum principle
ball.p = ball.p + (Fspring + Fgravity) * deltat
</syntaxhighlight>


Work done by a nonconstant force is more commonly seen in every day life than work done by a constant force. You can tell if a force is nonconstant if the object moves a distance with a changing force at points along the path. Two examples of nonconstant forces are spring forces and gravitational forces. You can tell that a gravitational force is a nonconstant force by choosing a point on the path. For example, if you choose a point to calculate force on the Moon's orbit around Earth then the Moon will go away from the Earth rather than orbit around it. Another example of a nonconstant force is a spring. If a spring had a constant force, the spring would forever stretch or compress rather than oscillate.


To calculate a nonconstant force you must use a different formula than W=F*d. An integral is needed to calculate the work done along a path of nonconstant force.
== Interactive Model ==
Try out this Trinket simulation of spring motion: 
[https://trinket.io/glowscript/49f7c0f35f View the simulation on Trinket]


[[File:WorkIntegral.png]]
== Examples ==
=== Simple ===
'''Question:''' 
A box is pushed 10 m east by a 40 N force, then 8 m north by a 60 N force.
'''Solution:''' 
: <math>W = 40 \cdot 10 + 60 \cdot 8 = 880 \, J</math>


=== Middling ===
'''Question:''' 
A spring with <math>k = 70 \, N/m</math> is stretched 10 cm. 
[[File:Middle1.JPG|thumb|center|200px|Spring stretching setup]]


===Mathematical Model===
'''Solution:''' 
: <math>W = \frac{1}{2} k x^2 = \frac{1}{2}(70)(0.1)^2 = 0.35 \, J</math>


=== Difficult ===
'''Question:''' 
How much work is done by Earth’s gravity on an asteroid falling from distance <math>d</math> to radius <math>R</math>?


[[File:particle.jpg|thumb|Particle with nonconstant force]]
'''Solution:''' 
Start with Newton’s law of gravitation:
: <math>F = \frac{GMm}{r^2}</math>


The total amount of work done on a system is equal to the sum of the work done by all individual forces, therefore, the total amount of work done can be calculated by the summation of each force multiplied by the force
Then integrate:
[[Iterative Prediction of Spring-Mass System|Iterative calculations]] are used in order to calculate non-constant forces and predict an object's motion. Given initial and final states of a system under non-constant force, small displacement intervals should be used to calculate the object's trajectory.
: <math>W = \int_R^d \frac{GMm}{r^2} \, dr = GMm \left( \frac{1}{R} - \frac{1}{d} \right)</math>


Below is the formula used to calculate each iteration then add them together to calculate total work.
== Connectedness ==
Understanding work by nonconstant forces is key in many fields:


<math>{{W}_{total} = {W}_{1} + {W}_{2} + {W}_{3} + ... + {W}_{n} = \overrightarrow{F}_{1}\bullet\overrightarrow{dr}_{1} + \overrightarrow{F}_{2}\bullet\overrightarrow{dr}_{2} + \overrightarrow{F}_{3}\bullet\overrightarrow{dr}_{3} + ... + \overrightarrow{F}_{n}\bullet\overrightarrow{dr}_{n}}</math>
* '''Springs''': Used in trampolines, shock absorbers, and mechanical pens 
* '''Engineering''': Fluid tanks fill unevenly, requiring nonconstant work 
* '''Energy''': Hydroelectric turbines rely on variable water flow 
* '''Space physics''': Rockets and satellites feel variable gravity


<math>{{W}_{total} = \sum\overrightarrow{F}\bullet\Delta\overrightarrow{r}}</math>


This method, while possible, can get tedious and repetitive. If you make the intervals you calculate indefinitely small, it is the same as integrating. The most common formula used for work with a nonconstant force is the integral from the first point of a path to the last point.  
== History ==
Gaspard-Gustave de Coriolis was the first to define "work" as force over distance. Later physicists used calculus to model work by nonconstant forces.


<math>{{W}_{total} = \int\limits_{i}^{f}\overrightarrow{F}\bullet\overrightarrow{dr}}</math>
== Further Reading & External Links ==


Here is a graphical example of the integral.  
=== Book ===
* Chabay & Sherwood – ''Matter and Interactions'' (4th ed.)


[[File:Graphs.png]]
=== Articles ===
* [https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/13%3A_Energy_Kinetic_Energy_and_Work/13.05%3A_Work_done_by_Non-Constant_Forces Nonconstant Force]
* [https://trinket.io/glowscript/49f7c0f35f Iterative Spring-Mass Simulation]


===Computational Model===
=== Simulations ===
 
* [https://trinket.io/glowscript/49f7c0f35f Spring-Mass Trinket Model]
[[File:Spring-Mass Motion along an axis.jpg|thumb|Spring-Mass Motion along an axis]]
[<https://trinket.io/glowscript/49f7c0f35f> Model of an Oscilating Spring]
 
This model shows both the total work and the work done by a spring on a ball attached to a vertical spring. The work done by the spring oscillates because the work is negative when the ball is moving away from the resting state and is positive when the ball moves towards it.
 
Because gravity causes the ball’s minimum position to be further from the spring’s resting length than its maximum position could be, the work is more negative when the ball approaches its minimum height.
 
The code works by using small time steps of 0.01 seconds and finding the work done in each time step. Work is the summation of all of the work done in each time step, so another step makes sure the value for work is cumulative.
 
 
====Modeling Non-Constant Forces in VPython====
 
[[File:Spring-Mass Motion in a 2-D plane.jpg|thumb|Spring-Mass Motion in a 2-D plane]]
As shown in this trinket model, [https://trinket.io/glowscript/c26c4c2637 Planer Motion of a Spring-Mass System], computational models can also be used in predicting non-constant forces in multiple directions.
 
#intialize conditions
#calculation loop
  #calculate/update force at every time step
    L = ball.pos - spring.pos
    Lhat = norm(L)
    s = mag(L) - L0
    Fspring = -(ks * s) * Lhat
  #apply momentum principle
    ball.p = ball.p + (Fspring + Fgravity) * deltat
  #update positions
  #update time
 
==Examples==
 
===Simple===
 
Situation
A box is pushed to the East 10 meters by a force of 40 N, then it is pushed to the north 8 meters by a force of 60 N.
Calculate the total work done on the box.
 
Diagram
 
[[File:simple1.jpg]]
 
Solution
 
<math> W = \sum\overrightarrow{F}\bullet\Delta\overrightarrow{r} </math>
 
<math> W = 40N \bullet\ 10m + 60N \bullet\ 8m </math>
 
<math> W = 40N \bullet\ 10m + 60N \bullet\ 8m </math>
 
<math> W = 880 J </math>
 
===Middling===
 
Background
We know that the formula for force is <math> F=ks </math>, where <math> s </math> is the distance the spring is stretched.
If we integrate this with respect to <math> s </math>, we find that <math> W=.5ks^2 </math> is the formula for work.
 
<math> W=\int\limits_{i}^{f}\overrightarrow{k}\bullet\overrightarrow{ds} = .5ks^2 </math>
 
Situation
 
Say that we want to find the work done by a horizontal spring with spring constant k=70 N/m as it moves an object 10 cm.
 
Diagram
 
[[File:Middle1.jpg]]
 
 
Solution
 
Using the formula W=.5ks2 that we derived from F=ks, we can calculate that the work done by the spring is  0.35 J.
 
<math> W=\int\limits_{0}^{10}70\bullet\overrightarrow{ds}=.5ks^2=.5(70)(0.10^2)=0.35 J </math>
 
===Difficult===
The earth does work on an asteroid approaching from an initial distance r. How much work is done on the asteroid by gravity before it hits the earth’s surface?
 
First, we must recall the formula for gravitational force.
 
Because <math> G </math>, <math> M </math>, and <math> m </math> are constants, we can remove them from the integral. We also know that the integral of <math> -1\over r^2 </math> is <math> 1\over r </math>. We then must calculate the integral of <math> –GMm\over r^2 </math> from the initial radius of the asteroid, <math> R </math>, to the radius of the earth, <math> r </math>.
 
<math> W=-GMm\bullet\int\limits_{R}^{r}{-1\over r^2}\bullet dr </math>
 
<math> W=-GMm\bullet({1\over r}-{1\over R}) </math>
 
Our answer will be positive because the forces done by the earth on the asteroid and the direction of the asteroid's displacement are the same.
 
==Connectedness==
I am most interested in the types of physics problems that accurately model real world situations. Some forces, like gravity near the surface of the earth and some machine-applied forces, are constant. However, most forces in the real world are not.
 
Because of this, calculating work for non-constant forces is essential to mechanical engineering. For example, when calculating work done by an engine over a distance, the force applied by the engine can vary depending on factors such as user controls.
 
On an industrial level, the work needed to fill empty tanks depends on the weight of the liquid, which varies as the tanks fill and empty. Energy conversion in hydroelectric dams depends on the work done by water against turbines, which depends on the flow of water. Windmills work in the same way.
 
 
==History==
 
Gaspard-Gustave de Coriolis, famous for discoveries such as the Coriolis effect, is credited with naming the term “work” to define force applied over a distance. Later physicists combined this concept with Newtonian calculus to find work for non-constant forces.
 
 
== See also ==
 
'''Further Reading'''
 
[[Work]]
 
[[Iterative Prediction of Spring-Mass System]]
 
 
'''External Links'''
 
https://www.youtube.com/watch?v=jTkknXVjBl4
 
https://www.youtube.com/watch?v=9Be81qfgBVc
 
==References==
 
[http://www.britannica.com/biography/Gustave-Gaspard-Coriolis]
[http://www.math.northwestern.edu]
 
[[Category:Energy]]
 
Created by Justin Vuong
 
Edited by Chris Mickas
 
Edited by Yunqing Jia

Latest revision as of 22:24, 22 April 2025

Work Done By A Nonconstant Force

Claimed by Matt McCrory – Spring 2025

This page explains how to calculate work done when the force applied is not constant. It includes conceptual explanations, worked examples, mathematical and computational models, and embedded simulations to make this concept easier to understand.


The Main Idea

Before we understand nonconstant force, let's review constant force.

For constant force:

Work = Force × Distance
[math]\displaystyle{ W = F \cdot d }[/math]
Work as the area under a constant force graph

In real-life, however, forces often vary over distance. In that case, we use:

[math]\displaystyle{ W = \int_{x_1}^{x_2} F(x) \, dx }[/math]

This integral calculates the total work as the area under the curve on a Force vs. Distance graph.

Mathematical Model

Work done by a varying force is found by breaking the motion into tiny intervals:

[math]\displaystyle{ W = \sum \vec{F}_i \cdot \Delta \vec{r}_i }[/math]

As the interval becomes very small, it becomes a definite integral:

[math]\displaystyle{ W = \int \vec{F} \cdot d\vec{r} }[/math]

Spring Example

If [math]\displaystyle{ F = kx }[/math], we derive:

[math]\displaystyle{ W = \int_0^x kx \, dx = \frac{1}{2}kx^2 }[/math]
Work done by a spring force

Computational Model

Computational models can approximate work using many tiny time steps. Below is Python code modeling a vertical spring in VPython:

<syntaxhighlight lang="python">

  1. initialize conditions

L = ball.pos - spring.pos Lhat = norm(L) s = mag(L) - L0 Fspring = -(ks * s) * Lhat

  1. momentum principle

ball.p = ball.p + (Fspring + Fgravity) * deltat </syntaxhighlight>


Interactive Model

Try out this Trinket simulation of spring motion: View the simulation on Trinket

Examples

Simple

Question: A box is pushed 10 m east by a 40 N force, then 8 m north by a 60 N force. Solution:

[math]\displaystyle{ W = 40 \cdot 10 + 60 \cdot 8 = 880 \, J }[/math]

Middling

Question: A spring with [math]\displaystyle{ k = 70 \, N/m }[/math] is stretched 10 cm.

Spring stretching setup

Solution:

[math]\displaystyle{ W = \frac{1}{2} k x^2 = \frac{1}{2}(70)(0.1)^2 = 0.35 \, J }[/math]

Difficult

Question: How much work is done by Earth’s gravity on an asteroid falling from distance [math]\displaystyle{ d }[/math] to radius [math]\displaystyle{ R }[/math]?

Solution: Start with Newton’s law of gravitation:

[math]\displaystyle{ F = \frac{GMm}{r^2} }[/math]

Then integrate:

[math]\displaystyle{ W = \int_R^d \frac{GMm}{r^2} \, dr = GMm \left( \frac{1}{R} - \frac{1}{d} \right) }[/math]

Connectedness

Understanding work by nonconstant forces is key in many fields:

  • Springs: Used in trampolines, shock absorbers, and mechanical pens
  • Engineering: Fluid tanks fill unevenly, requiring nonconstant work
  • Energy: Hydroelectric turbines rely on variable water flow
  • Space physics: Rockets and satellites feel variable gravity


History

Gaspard-Gustave de Coriolis was the first to define "work" as force over distance. Later physicists used calculus to model work by nonconstant forces.

Further Reading & External Links

Book

  • Chabay & Sherwood – Matter and Interactions (4th ed.)

Articles

Simulations