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:
As the interval becomes very small, it becomes a definite integral:
: <math>W = \int \vec{F} \cdot d\vec{r}</math>
Work done by a constant force is 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.
While this formula is useful, it is not realistic to assume force will be constant in every system.
[[File:WorkIntegral.png|thumb|center|300px|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:
'''Work done by a Nonconstant Force'''
<syntaxhighlight lang="python">
#initialize conditions
L = ball.pos - spring.pos
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 gravitational force is a nonconstant force by observing the Moon's orbit and comparing different points on its path. If you choose points at different distances from the Earth to calculate the gravitational force on the Moon, you would observe that the gravitational force is greater at a closer distance from the Earth. 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.
Lhat = norm(L)
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, where work is the is change in energy of a system by a force.
s = mag(L) - L0
[[File:WorkIntegral.png]]
Fspring = -(ks * s) * Lhat
#momentum principle
ball.p = ball.p + (Fspring + Fgravity) * deltat
</syntaxhighlight>
===Mathematical Model===
== Interactive Model ==
Try out this Trinket simulation of spring motion:
[https://trinket.io/glowscript/49f7c0f35f View the simulation on Trinket]
== Examples ==
=== Simple ===
'''Question:'''
A person pushes a box and slowly loses strength after ten seconds. It can be described using this formula: F(x) = 20 - 2x. How much work was applied to the box?
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 distance.
'''Solution:'''
[[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.
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.
== 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
Here is a graphical example of the integral.
An interesting industrial example of work done by a non-constant force is the operation of a car engine. The torque produced by an internal-combustion engine is not constant, it depends on the engine speed (RPM). As the RPM changes, the engine’s torque follows a characteristic “torque curve,” rising to a peak and then decreasing. Because the driving force on the wheels is proportional to engine torque, the force applied to the car varies continuously, making it a real-world industrial example of work done by a non-constant force.
[[File:Graphes.png]]
== 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.
===Computational Model===
== Further Reading & External Links ==
[[File:Spring-Mass Motion along an axis.jpg|thumb|Spring-Mass Motion along an axis]]
=== Book ===
[<https://trinket.io/glowscript/49f7c0f35f> Model of an Oscilating Spring]
* Chabay & Sherwood – ''Matter and Interactions'' (4th ed.)
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.
[[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===
'''Question'''
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.
[[File:Simple1.JPG]]
'''Solution'''
<math> W = \sum\overrightarrow{F}\bullet\Delta\overrightarrow{r} </math>
The earth does work on an asteroid approaching from an initial distance d.
How much work is done on the asteroid by gravity before it hits the earth’s surface?
[[File:Diff1.JPG]]
'''Solution'''
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 d^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 d^2}\bullet dr </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==
The sector of physics that interests me the most is when I can see what I am computing. This unit covers spring forces. I can understand how the force is not constant because I can see the spring oscillating and changing positions and the amount of force.
Work done by a nonconstant force is related to my major, Industrial Engineering, because my major focuses heavily on optimization. Understanding how nonconstant forces work will help with machines in the workplace to help optimize efficiency.
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'''
Book:
Matter and Interactions - 4e Chabay and Sherwood
Article:
[[http://physics.tutorvista.com/forces/non-contact-force.html Non Constant Force]]
[[Iterative Prediction of Spring-Mass System]]
'''External Links'''
[[https://www.youtube.com/watch?v=jTkknXVjBl4 Work done by Non Constant Force]]
[[https://www.youtube.com/watch?v=9Be81qfgBVc Work done by Constant Force]]
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:
Question:
A person pushes a box and slowly loses strength after ten seconds. It can be described using this formula: F(x) = 20 - 2x. How much work was applied to the box?
Solution:
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
An interesting industrial example of work done by a non-constant force is the operation of a car engine. The torque produced by an internal-combustion engine is not constant, it depends on the engine speed (RPM). As the RPM changes, the engine’s torque follows a characteristic “torque curve,” rising to a peak and then decreasing. Because the driving force on the wheels is proportional to engine torque, the force applied to the car varies continuously, making it a real-world industrial example of work done by a non-constant force.
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.)
