Work Done By A Nonconstant Force: Difference between revisions

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This page will help students understand how to calculate the work done by a non constant force.
= Work Done By A Nonconstant Force =
'''Claimed by Matt McCrory – Spring 2025'''


==The Main Idea==
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.


When calculating the force, if the magnitude of the force or direction of the force changes, it is not possible to calculate the work done by multiplying force by the displacement. Instead the non constant force is split into a path with small increments.


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


===A Mathematical Model===
For constant force:
: '''Work = Force × Distance'''
: <math>W = F \cdot d</math>


<math> W=\int\limits_{i}^{f}\overrightarrow{F}\bullet\overrightarrow{dr} = \sum\overrightarrow{F}\bullet\Delta\overrightarrow{r} </math>
[[File:ConstantForce.png|thumb|center|300px|Work as the area under a constant force graph]]


This means that the work is equal to the integral of the function of the force with respect to the change in the objects position. This is also the same as the summation of the force on an object multiplied by the change in position.
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>


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


[[File:Screen Shot 2015-12-05 at 5.23.32 PM.png]]
== Mathematical Model ==
Work done by a varying force is found by breaking the motion into tiny intervals:


This python code creates a ball with a force acting on it that changes with respect to time and it prints the total work at the end of the the loop that lasts while t is less than 10.
: <math>W = \sum \vec{F}_i \cdot \Delta \vec{r}_i</math>
This uses the concept that work is equal to the summation of the force multiplied by the change in distance over that interval, which is an estimate for the integral of the force function over this distance.


==Examples==
As the interval becomes very small, it becomes a definite integral:
: <math>W = \int \vec{F} \cdot d\vec{r}</math>


===Example 1===
=== Spring Example ===
A box is pushed to the East, 5 meters by a force of 40 N, then it is pushed to the north 7 meters by a force of 60 N. Calculate the work done on the box.
If <math>F = kx</math>, we derive:
: <math>W = \int_0^x kx \, dx = \frac{1}{2}kx^2</math>


<math> W = \sum\overrightarrow{F}\bullet\Delta\overrightarrow{r} </math>
[[File:WorkIntegral.png|thumb|center|300px|Work done by a spring force]]


<math> W = 40N \bullet\ 5m + 60N \bullet\ 7m </math>
== Computational Model ==
Computational models can approximate work using many tiny time steps. Below is Python code modeling a vertical spring in VPython:


<math> W = 40N \bullet\ 5m + 60N \bullet\ 7m </math>
<syntaxhighlight lang="python">
#initialize conditions
L = ball.pos - spring.pos
Lhat = norm(L)
s = mag(L) - L0
Fspring = -(ks * s) * Lhat


<math> W = 620 J </math>
#momentum principle
ball.p = ball.p + (Fspring + Fgravity) * deltat
</syntaxhighlight>


===Example 2===
As a ball is attached to a spring and moves to the right. The ball moves 5 meters to the right and the spring constant of the spring is 5 N/m. How much work is done by the spring?


<math> W=\int\limits_{i}^{f}\overrightarrow{F}\bullet\overrightarrow{dr} </math>
== Interactive Model ==
Try out this Trinket simulation of spring motion: 
[https://trinket.io/glowscript/49f7c0f35f View the simulation on Trinket]


<math> F = -k \bullet\ r </math>
== 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>


<math> W=\int\limits_{0}^{5m} -k \bullet\ dr </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]]


<math> W=\int\limits_{0}^{5m} -5 \bullet\ dr </math>
'''Solution:''' 
: <math>W = \frac{1}{2} k x^2 = \frac{1}{2}(70)(0.1)^2 = 0.35 \, J</math>


<math> W=\int\limits_{0}^{5m} -5 \bullet\ dr </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>?


<math> W=-5 ((5m^2)/2 - 0) </math>
'''Solution:''' 
Start with Newton’s law of gravitation:
: <math>F = \frac{GMm}{r^2}</math>


<math> W= 62.5 J </math>
Then integrate:
: <math>W = \int_R^d \frac{GMm}{r^2} \, dr = GMm \left( \frac{1}{R} - \frac{1}{d} \right)</math>


==Connectedness==
== Connectedness ==
How is this topic connected to something that you are interested in?
Understanding work by nonconstant forces is key in many fields:


Even though I'm an ECE major, I have an interest in aviation, and the force of a jet engine is not always a constant force, so you would need to use this method to calculate the work done instead of the simple method.
* '''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


How is it connected to your major?


As an ECE major, this could be connected by my major when working with an electric motor and calculating the amount of power needed to power the motor.
== 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.


Is there an interesting industrial application?
== Further Reading & External Links ==


Since many forces in the real world are not constant, this method of calculating work is needed for most situations.
=== Book ===
* Chabay & Sherwood – ''Matter and Interactions'' (4th ed.)


==History==
=== 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]


The concept of work was introduced by a French mathematician named Gaspard-Gustave Coriolis in 1826. The concept was established as a "weight lifted through a height".
=== Simulations ===
 
* [https://trinket.io/glowscript/49f7c0f35f Spring-Mass Trinket Model]
== See also ==
 
[[Work]]
 
==References==
 
[http://www.britannica.com/biography/Gustave-Gaspard-Coriolis]
[https://en.wikibooks.org/wiki/FHSST_Physics/Work_and_Energy/Work]
 
[[Category:Energy]]
 
Claimed By Justin V.

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