Potential energy for a spring law

Click For Summary

Discussion Overview

The discussion revolves around the derivation and understanding of the potential energy formula for springs, specifically the equation Ep = 1/2 k x^2. Participants explore the origins of this formula, its implications, and the differences between various expressions of work and energy related to springs.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant seeks clarification on the formula Ep = 1/2 k x^2, questioning the meaning of x^2 and the reason for the coefficient of 1/2.
  • Another participant explains that the force from a spring is given by k*x and that energy is the integral of this force with respect to distance, leading to the derivation of the potential energy formula.
  • A non-calculus explanation is provided, stating that the work done on a spring is not constant and varies with compression, which results in the potential energy being expressed as Ep = 1/2 k x^2.
  • Participants discuss the concept of average force in the context of non-constant force during spring compression, leading to the conclusion that the energy input is half of the maximum force times the distance.
  • Questions are raised about the value of potential energy when the spring is not compressed and how this relates to the change in potential energy during compression.
  • Some participants note the relevance of the factor of 1/2 in various physics contexts, suggesting a broader mathematical principle at play.
  • There are mentions of challenges with LaTeX formatting for equations, indicating a technical aspect of the discussion.
  • One participant expresses frustration with applying formulas to problems despite understanding them in theory, highlighting a common struggle in learning physics.

Areas of Agreement / Disagreement

Participants generally agree on the derivation of the potential energy formula and its implications, but there are ongoing questions and clarifications regarding specific aspects of the formula and its application. The discussion remains somewhat unresolved regarding the deeper understanding of potential energy changes and the participant's personal challenges with physics problems.

Contextual Notes

Some limitations include the dependence on understanding calculus for the derivation of the potential energy formula and the varying interpretations of the formula's application in different contexts. Additionally, the discussion reflects a range of experiences and levels of understanding among participants.

Ward_edojawa
Messages
15
Reaction score
0
Hi guys! Thanks for this great forum. I have a question I need explanation about please...
I just started 11th grade and it's my first time taking springs. In the book, they said :
Ep= 1/2 k x^2
I already understand all the previous laws like [w= -1/2 k (x2^2 _ x1^2)] and know where thay came from
But THIS law is not explained in the book and I didn't get where it came from...
What's x^2 ?? Why 1/2 and not -1/2??
Can u please help me with this??
Thank you :)
 
Physics news on Phys.org
The force from a spring is given by k*x
Energy is the integral of the force with respect to distance (x in this case)

So we have ∫kx dx=1/2 kx^2
 
  • Like
Likes   Reactions: 1 person
Potential energy is an amount of energy available to do work.

You said you understand this formula
w= -1/2 k (x2^2 _ x1^2)
which describes the change in potential energy as the spring elongation (or compression) changes, or W = Ep(2) -Ep(1).

A description not involving calculus
--------------------------------------
You should know that when you push an object by a force (F) a distance (x), the work you have done W = Fx. You could say that some amount of potential energy that you once had, has now been used to do work. This previous amount of potential energy is Ep = W.

For a spring, the force is not constant but varies from a value of 0 at free extension to a value of F at compression.

So, draw yourself a graph with force as the ordinate and distance the spring is compressed as the abscissa.
The graph of the spring compression should be a straight line starting at (0,0) with slope k, and at the final compression of the spring the coordinates of force and compression are (x,F).

Constant force:
If the force to compress the spring had been constant the energy put into the spring would have been the work done by you or,
Ep = Fx
and obviously that is not the energy input into a spring during compression.

Non-constant force:
With a non-constant force during compression, but one that is proportional to distance of compression, you can see that the energy input is Fx/2, or
Ep = Fx/2

Knowing that F = kx, a substitution evaluates to
Ep = 1/2 kx^2
 
  • Like
Likes   Reactions: Ward_edojawa
Why is it 1/2 and not -1/2.
Potential energy of the spring has what value when it is not compressed?
What is the potential energy with compression?
What is the change in potential energy?
 
  • Like
Likes   Reactions: 1 person
Non-calculus explanation. W=f*d, but f varies linearly with d. So average force is half of peak force (force at full d).
 
  • Like
Likes   Reactions: 1 person
256bits said:
Potential energy is an amount of energy available to do work.

You said you understand this formula

which describes the change in potential energy as the spring elongation (or compression) changes, or W = Ep(2) -Ep(1).

A description not involving calculus
--------------------------------------
You should know that when you push an object by a force (F) a distance (x), the work you have done W = Fx. You could say that some amount of potential energy that you once had, has now been used to do work. This previous amount of potential energy is Ep = W.

For a spring, the force is not constant but varies from a value of 0 at free extension to a value of F at compression.

So, draw yourself a graph with force as the ordinate and distance the spring is compressed as the abscissa.
The graph of the spring compression should be a straight line starting at (0,0) with slope k, and at the final compression of the spring the coordinates of force and compression are (x,F).

Constant force:
If the force to compress the spring had been constant the energy put into the spring would have been the work done by you or,
Ep = Fx
and obviously that is not the energy input into a spring during compression.

Non-constant force:
With a non-constant force during compression, but one that is proportional to distance of compression, you can see that the energy input is Fx/2, or
Ep = Fx/2

Knowing that F = kx, a substitution evaluates to
Ep = 1/2 kx^2




Well THANK YOU SOOOOOOOOOOOOOOOOOOOOOOOOOOO MUCH! :!) :thumbs: I get it now! I really thank you :-D
And about -1/2 I tried getting the formula of the potential energy from the ( wfs = -1/2 k (x2^2 - x1^2) )
But now I get it. It's my first time taking the lesson so excuse my stupidity :shy: :blushing: thanks again ^.^
 
Thank you guys! I get it now! Thanks to all of you ;) <3
 
256bits said:
So, draw yourself a graph with force as the ordinate and distance the spring is compressed as the abscissa. The graph of the spring compression should be a straight line starting at (0,0) with slope k, and at the final compression of the spring the coordinates of force and compression are (x,F).

Note the "area" of a region on this graph has units of force times distance = work. The work done in compressing the spring is the "area" of the triangular region below the graph that 256bits decribed. What's the area of a triangle with width x and height F?

Now substitute F = kx into that...
 
  • Like
Likes   Reactions: 1 person
This same factor of a half comes into the calculation for distance traveled in time t, under acceleration a :
s = at2/2
and in many other instances, like the energy stored in a Capacitor:
E = CV2/2
It's the same basic bit of Calculus that produces the same factor, all over Physics.
 
  • Like
Likes   Reactions: Ward_edojawa
  • #10
I obviously need help with LaTex. The following looks OK on my preview but comes out in its original form when I post it.
Thus
$$ E = \frac{C \ V^2}{2} $$

Wassup?
 
  • #11
sophiecentaur said:
I obviously need help with LaTex. The following looks OK on my preview but comes out in its original form when I post it.
Thus
$$ E = \frac{C \ V^2}{2} $$

its okay I get it with normal writings XD and thank you ! :)
 
  • #12
That problem was crazy. I can now see the formula just as i want it. I may have been impatient at first. It seems to take a while to cope with the tex rendering.

But, back to the question and my comment - there are so many times where a formula fits into different contexts with different variables. Just cos there's a smartypants answer for why, doesn't make it any less remarkable. :)
 
  • #13
sophiecentaur said:
That problem was crazy. I can now see the formula just as i want it. I may have been impatient at first. It seems to take a while to cope with the tex rendering.

But, back to the question and my comment - there are so many times where a formula fits into different contexts with different variables. Just cos there's a smartypants answer for why, doesn't make it any less remarkable. :)
well i have a serious problem with physics... i study all formulas and understand them and get examples and all... then i put a problem and i just like o_O ... like PLZ OMG I UNDERSTAND EVERYTHING! :( if I don't get full a full mark in physics next year (12th grade) and this current year (11th grade) all my life will fall apart...
 

Similar threads

Undergrad Question about potential energy: gravity and a vertical spring
  • · Replies 12 ·
Replies
12
Views
4K
Undergrad Work-Energy Theorem for Moving Block and Spring System?
  • · Replies 4 ·
Replies
4
Views
3K
High School How will a spring stretch in the following cases?
  • · Replies 17 ·
Replies
17
Views
2K
Undergrad Is Hooke's law really not working, or is it me being dummy?
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Change of coordinates in a potential energy field
  • · Replies 2 ·
Replies
2
Views
2K
High School Why is my experimental spring constant an order of magnitude off?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Potential ##V## and potential energy ##E_{pot}##?
  • · Replies 2 ·
Replies
2
Views
2K
High School Conceptually understanding change in potential energy with 0 net work
  • · Replies 9 ·
Replies
9
Views
3K
High School Why Does Hooke's Law Use a Negative Sign in Scalar Form?
  • · Replies 30 ·
2
Replies
30
Views
4K
High School Curious about Work and Energy Consumption
  • · Replies 8 ·
Replies
8
Views
2K