Potential Energy of Macroscopic Springs: Difference between revisions

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Claimed by Hailey Lindstrom for Spring 2017
Claimed by Hailey Lindstrom for Spring 2017


==The Key Concept==
==The Main Idea==


A macroscopic ideal spring is an elastic device which stores potential energy when stretched or compressed. Ideal springs exert a force which is linearly proportional to its change in length. This relationship can be modeled by the equation '''F = -kx''', where '''F''' is the force, '''K''' is the spring constant in N/m or lb/in, and '''x''' is the change in length of the spring from its equilibrium length.
A macroscopic ideal spring is an elastic device which stores potential energy when stretched or compressed. Ideal springs exert a force which is linearly proportional to its change in length. This relationship can be modeled by the equation '''F = -kx''', where '''F''' is the force, '''k''' is the spring constant in N/m or lb/in, and '''x''' is the change in length of the spring from its equilibrium length.
 
==Terms and Definitions==


===A Mathematical Model===
===A Mathematical Model===


'''Potential Energy''' - the energy stored in a body or system that results inherently from its relative position to another object
The equation for potential energy of a spring can be mathematically shown by integrating Hooke's Law.  
* A change in potential energy occurs when there is a change in the separation distance between the two objects.
* Gravitational potential energy relates the vertical position of an object to the surface of the Earth.
<math> \Delta U\approx \Delta mgy </math>


Lanier begins here and continues thru computational section:
* We start with the initial equation for Force of a macroscopic spring, which is equal to:


* With spring potential energy, we relate potential energy to one half the product of the spring constant and the square of equilibrium displacement, or this:
<math> F = kx </math>


<math> \Delta U_{spring} = \frac{1}{2}k\Delta x^2 </math>
* We then integrate with respect to x so that we can obtain work (which is equal to the integral of force):


* derive this, consider the following from the work-energy theorem:
<math>\int F\  dx = \int kx\ dx </math>


<math> \Delta U_{spring} = W </math>
* Finally, we solve the indefinite integral which gives us the final equation for work of a macroscopic spring.


*We then look at the definition of work and infer the following:
<math> W = \frac{1}{2}kx^2 </math>


<math> W = F\Delta x </math>
Here, work is equal to the elastic spring potential energy, <math> \Delta U_{spring}</math>.


<math> \Delta U_{spring} = f\Delta x </math>
===A Computational Model===


Using the mean value theorem of integrals, we simplify the nonconstant force using the mean value theorem:
Below is a graph of force versus displacement for a macroscopic spring with different spring constants (k values). The x axis is displacement in meters and the y axis is in Newtons. This shows that there is a linear relationship between change in length and force.


<math> F_{avg} = \frac{k}{\Delta x}\int_{x_1}^{x_2}x \;dx </math>


<math> F_{avg} = \frac{k}{\Delta x}\cdot \frac{\Delta x^2}{2} </math>
[[File:finalfinalgraph.png]]


<math> F_{avg} = \frac{k\Delta x}{2} </math>
Finally, we arrive at the above formula for spring potential energy:
<math> \Delta U_{spring} = \frac{k\Delta x}{2}\cdot \Delta x </math>
<math> \Delta U_{spring} = \frac{1}{2}k\Delta x^2 </math>
Ta-dah!
===A Computational Model===


When we plot the change in potential energy of a spring as a function of equilibrium displacement in vPython, we arrive at a parabolic model like the one in this image:


[[File:Parabolicmodel.jpg]]
Represented here is the potential energy of the macroscopic spring as a function of the change in length of the spring. The x axis is displacement in meters and the y axis is energy in Joules. This shows that the relationship between change in length and potential energy is quadratic.  


Our independent variable is the displacement of the spring from its equilibrium position, and the dependent variable is the product of the displacement squared and <math> \frac{k}{2} </math>.
[[File:quadraticgraphhailey.png]]


==Examples==
==Examples==


All Lanier
===Simple===


'''Simple'''
'''Question'''


''If a spring has a spring constant of <math> 1000 \;N/m </math> and is stretched <math> 10 \;cm </math>, what is its potential energy?''
If a spring has a spring constant, k = 1000 N/m, and is stretched 10 cm, find the force and potential energy of the spring.


<math> U = \frac{1}{2}(1000)(0.1)^2 </math>
'''Solution'''


<math> U = 5\;J </math>
[[File:lastexampleofthis.jpg]]


'''Moderate'''
===Moderate===


''If a spring has a spring constant of <math> 1000 \;N/m </math> and is stretched <math> 10 \;cm </math> by an unknown mass, find the unknown mass.''
'''Question'''


<math> F = k\Delta x </math>
If a spring has a spring constant, k = 1000 N/m and is stretched 10 cm by an unknown mass, find the unknown mass.


<math> m = \frac{k\Delta x}{g} </math>
'''Solution'''


<math> m = \frac{(1000)(0.1)}{9.8} </math>
[[File:haileyexample1-2.jpg]]


<math> m \approx 10.204\;kg </math>
===Difficult===


'''Difficult'''
'''Question'''


''If the total potential energy of a system consisting of a mass suspended on a spring is <math> 100\;J</math>, the mass is two metres off the ground,  and the mass of the object is <math>4.081\;kg</math>, find the spring potential energy.''
If there is a cart of mass, m = 1 kg, moving at speed, v = 10 m/s^2, and it collides with a spring of spring constant, k = 1000 N/m, how far does the spring compress? You can ignore friction in this example!


<math> \sum U = U_{g}+U_s </math>
'''Solution'''


<math> U_s = 100-mgh </math>
[[File:fullsizerender-2.jpg]]


<math> U_s = 100-(4.081)(9.8)(2) </math>
==Connectedness==
'''Hailey's Answers'''


<math> U_s = 20 J </math>
1. How is this topic connected to something that you are interested in?
The topic of macroscopic springs is actually much more interesting to me than I originally thought before creating this wiki page. There are so many practical applications of springs in the real world that make life much easier. For example, springs are used in cars, particularly in shock absorbers, and this helps us to drive more safely. Another use of springs that we all use is everyday is the springs within the mattress of our bed. Having the right spring stiffness (k value) is really important because it provides you with a great (or little) amount of support and comfort while sleeping.
2. How is it connected to your major?
This concept is related to my major in the sense that solving potential energy or force for a macroscopic spring requires a basic understanding of calculus - particularly integrals. In addition, some of the more difficult spring problems require a higher understanding of mathematic equations as they relate to something else (in this case, it is physics). This skill is something that is useful in industrial engineering because it deals with a lot of applying mathematic equations to different concepts.


==Connectedness==
3. Is there an interesting industrial application?
#How is this topic connected to something that you are interested in?
As stated above, springs are used a lot in the automobile industry. One thing that I found really interesting was that the use of springs in vehicles actually allows for massive amounts of weight to be supported. They are able to work with the shock absorbers to help absorb larger weights than the vehicle on its own would be able to support. They also help in flexibility of automobiles which helps to provide for a smoother ride.  
This is the first law of thermodynamics where every energy related goes around with this law, where energy is neither made or destroyed.  
It is very interesting how energy is just there and is transformed into other energies such as chemical energy that the food in the student center has will transform into kinetic energy when playing tennis after school.
#How is it connected to your major?
As my major is Chemical Engineering, thermodynamics has many materials in common because of calculating the energy balances toward a reaction. The first law of thermodynamics To work out thermodynamic problems you will need to isolate a certain portion of the universe, the system, from the remainder of the universe, the surroundings.


#Is there an interesting industrial application?
There was an interesting industrial application where we can calculate the energy required by the machine to pump the fluid out.


'''Lanier's Answers'''
'''Lanier's Answers'''
Line 111: Line 90:
2. This topic relates to being a math major for obvious reasons; it presents a basic, practical application for advanced mathematics, something that isn't seen often in my experience.
2. This topic relates to being a math major for obvious reasons; it presents a basic, practical application for advanced mathematics, something that isn't seen often in my experience.


3. A silly yet decent example of industrial applications of spring applications is the way many pens are constructed: A spring of specific dimensions requiring a spring constant within a certain range is necessary to make the button on a pen work.
3. A silly yet decent example of industrial applications of spring applications is the way many pens are constructed: a spring of specific dimensions requiring a spring constant within a certain range is necessary to make the button on a pen work.


==History==
==History==
Someone else:
Before the existence of coiled springs, leaf springs were used. These springs consisted of curved strips of metal clamped to one another. However, these were not very efficient because they always had to be lubricated and squeaked a lot. It was not until 1763 when R. Tradwell invented the first coiled spring. Finally in 1857 when a steel coil spring was invented. Springs progressed and within just a century, springs have been used in many different things and have substantially improved the car and machine industry.
 
*'''William Rankinet'''  
*'''William Rankinet'''  
**The term potential energy was introduced e, although it has links to Greek philosopher Aristotle's concept of potentiality.
**The term potential energy was introduced as e, although it has links to Greek philosopher Aristotle's concept of potentiality.
**Scottish engineer and physicist  
**Scottish engineer and physicist  
** links to Greek philosopher Aristotle's concept of potentiality


Lanier:
*'''Robert Hooke'''
*'''Robert Hooke'''
**Came up with Hooke's Law, which is used to find the potential energy of springs.
**Came up with Hooke's Law, which is used to find the potential energy of springs.
Line 126: Line 104:


== See also ==
== See also ==
Someone else:
Potential Energy
Ideal Spring
Spring stretch.


===Further reading===
===Further reading===


Broken link from someone else that I fixed
[http://en.wikipedia.org/wiki/Potential_energy Potential Energy]
[http://en.wikipedia.org/wiki/Potential_energy Potential Energy]
===External Links===
Good explanation (and calculator) for potential energy in macroscopic springs:
[http://hyperphysics.phy-astr.gsu.edu/hbase/pespr.html]


==References==
==References==


Someone else:
[http://hyperphysics.phy-astr.gsu.edu/hbase/pespr.html]
[https://www.generalspringkc.com/coil_springs_and_its_uses_s/2989.htm]
[http://coilingtech.com/history-springs/]
[https://en.wikipedia.org/wiki/William_John_Macquorn_Rankine]
[https://en.wikipedia.org/wiki/William_John_Macquorn_Rankine]
[https://en.wikipedia.org/wiki/Spring]
[https://en.wikipedia.org/wiki/Spring]
[https://en.wikipedia.org/wiki/Robert_Hooke]


Lanier:
[[Category:Energy]]
[https://en.wikipedia.org/wiki/Robert_Hooke]

Latest revision as of 18:35, 18 August 2019

An Exploration of Spring Potential Energy: Lanier Freeman and other


Claimed by Hailey Lindstrom for Spring 2017

The Main Idea

A macroscopic ideal spring is an elastic device which stores potential energy when stretched or compressed. Ideal springs exert a force which is linearly proportional to its change in length. This relationship can be modeled by the equation F = -kx, where F is the force, k is the spring constant in N/m or lb/in, and x is the change in length of the spring from its equilibrium length.

A Mathematical Model

The equation for potential energy of a spring can be mathematically shown by integrating Hooke's Law.

  • We start with the initial equation for Force of a macroscopic spring, which is equal to:

[math]\displaystyle{ F = kx }[/math]

  • We then integrate with respect to x so that we can obtain work (which is equal to the integral of force):

[math]\displaystyle{ \int F\ dx = \int kx\ dx }[/math]

  • Finally, we solve the indefinite integral which gives us the final equation for work of a macroscopic spring.

[math]\displaystyle{ W = \frac{1}{2}kx^2 }[/math]

Here, work is equal to the elastic spring potential energy, [math]\displaystyle{ \Delta U_{spring} }[/math].

A Computational Model

Below is a graph of force versus displacement for a macroscopic spring with different spring constants (k values). The x axis is displacement in meters and the y axis is in Newtons. This shows that there is a linear relationship between change in length and force.



Represented here is the potential energy of the macroscopic spring as a function of the change in length of the spring. The x axis is displacement in meters and the y axis is energy in Joules. This shows that the relationship between change in length and potential energy is quadratic.

Examples

Simple

Question

If a spring has a spring constant, k = 1000 N/m, and is stretched 10 cm, find the force and potential energy of the spring.

Solution

Moderate

Question

If a spring has a spring constant, k = 1000 N/m and is stretched 10 cm by an unknown mass, find the unknown mass.

Solution

Difficult

Question

If there is a cart of mass, m = 1 kg, moving at speed, v = 10 m/s^2, and it collides with a spring of spring constant, k = 1000 N/m, how far does the spring compress? You can ignore friction in this example!

Solution

Connectedness

Hailey's Answers

1. How is this topic connected to something that you are interested in? The topic of macroscopic springs is actually much more interesting to me than I originally thought before creating this wiki page. There are so many practical applications of springs in the real world that make life much easier. For example, springs are used in cars, particularly in shock absorbers, and this helps us to drive more safely. Another use of springs that we all use is everyday is the springs within the mattress of our bed. Having the right spring stiffness (k value) is really important because it provides you with a great (or little) amount of support and comfort while sleeping.

2. How is it connected to your major? This concept is related to my major in the sense that solving potential energy or force for a macroscopic spring requires a basic understanding of calculus - particularly integrals. In addition, some of the more difficult spring problems require a higher understanding of mathematic equations as they relate to something else (in this case, it is physics). This skill is something that is useful in industrial engineering because it deals with a lot of applying mathematic equations to different concepts.

3. Is there an interesting industrial application? As stated above, springs are used a lot in the automobile industry. One thing that I found really interesting was that the use of springs in vehicles actually allows for massive amounts of weight to be supported. They are able to work with the shock absorbers to help absorb larger weights than the vehicle on its own would be able to support. They also help in flexibility of automobiles which helps to provide for a smoother ride.


Lanier's Answers

1. This topic is of much interest to me because of the applications of the mean value theorem of calculus to problems dealing with potential energy--both spring and gravitational. I think this type of problem underscores the amazing power of the MVT.

2. This topic relates to being a math major for obvious reasons; it presents a basic, practical application for advanced mathematics, something that isn't seen often in my experience.

3. A silly yet decent example of industrial applications of spring applications is the way many pens are constructed: a spring of specific dimensions requiring a spring constant within a certain range is necessary to make the button on a pen work.

History

Before the existence of coiled springs, leaf springs were used. These springs consisted of curved strips of metal clamped to one another. However, these were not very efficient because they always had to be lubricated and squeaked a lot. It was not until 1763 when R. Tradwell invented the first coiled spring. Finally in 1857 when a steel coil spring was invented. Springs progressed and within just a century, springs have been used in many different things and have substantially improved the car and machine industry.

  • William Rankinet
    • The term potential energy was introduced as e, although it has links to Greek philosopher Aristotle's concept of potentiality.
    • Scottish engineer and physicist
  • Robert Hooke
    • Came up with Hooke's Law, which is used to find the potential energy of springs.
    • Coined the term "cell" in biology

See also

Further reading

Potential Energy

External Links

Good explanation (and calculator) for potential energy in macroscopic springs: [1]

References

[2] [3] [4] [5] [6] [7]

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