Work done to stretch a spring vs Work done by a spring?

In summary, the equation for work done to stretch a spring is W = 1/2k(x22-x12), while the work done by a spring on an object is W = 1/2k(x12-x22). This is because in the first case, it represents the work done on the spring by external forces, while in the second case, it represents the work done by the spring on the object. The displacement values are positioned differently due to the principle of action and reaction, where the forces acting on the deformations of the spring resist the external applied force. The sign of the work done also indicates whether energy is put in or taken out, with displacement being conventionally calculated as positive.
  • #1
lu6cifer
16
0
Why is it that the equation for work done to stretch a spring is
W = 1/2k(x22-x12)

while the work done by a spring on an object is
W = 1/2k(x12-x22) ?

That is, how come it's x22-x12 for the first case, and x12-x22 for the second?
 
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  • #2
In one case it's the work done on the spring and in the other its work done by the spring.
 
  • #3
No, I get the semantics part, that in one case it's work done on a spring and in another it's work that's being done by the spring, but what's the physics explanation for why the x's are positioned where they are?
 
  • #4
The works done are equal, but opposite.
 
  • #5
I realize that they're opposites, meaning one is a positive value and the other is negative, but how is that determined? That is, how would you derive that based on the laws of physics?
 
  • #6
lu6cifer said:
I realize that they're opposites, meaning one is a positive value and the other is negative, but how is that determined? That is, how would you derive that based on the laws of physics?

If a spring is stretched by a force, there are internal forces acting on the deformations of the spring, in a manner resisting the external applied force which stretches the spring, and their work is opposite of the work done by the external force. If the spring stretches, intuitively you could say that the particles in the spring tend to move apart, so between them there act forces which "try to move them back together". Fundamentally, it's the principle of action and reaction.
 
  • #7
lu6cifer said:
I realize that they're opposites, meaning one is a positive value and the other is negative, but how is that determined? That is, how would you derive that based on the laws of physics?
Displacement is calculated by convention to be positive.

Draw a set of axes. If you start at the origin and move to the right 5 units, your displacement is (x2-x1)=5-0=5. If you start at the origin and move to the left, your displacement is (x1-x2)=0-(-5)=5.
 
  • #8
I think that the confusion arises because x1 and x2 haven't actually been defined here. One x is the shorter length and the other is the longer length. It would be better to state that you put the 'start' value and 'finish' value in the same place in the formula every time. This will produce a sign for the value of work which tells you whether it's put in or got out.
The two quoted versions of the formulae are 'verbal' ways of indicating the sign of the answer.
I hope that hasn't just added confusion!
 
  • #9
The 'sign' of the work refers to whether you put energy in or get it out.
 
  • #10
Realize that you are replying to a thread almost 9 months old.
 
  • #11
Durr!
Will try harder next time.
 
  • #12
When an object does work on another object, it applies a force for a distance (W=Fd). Initially the work done or energy stored on a spring is (1/2)k(x1)^2 and the final is (1/2)k(x2)^2, which results in W=(1/2)k((x2)^2-(x1)^2). Now, when the spring releases, the initial position is x2 while the final position is x1, thus the order of the 2 terms in brackets become reversed. It can also be thought of as negative work done on the object. (-(a-b)=b-a) where a is x1 and b is x2
 

What is work done to stretch a spring?

Work done to stretch a spring refers to the amount of energy required to change the length of a spring from its original state to a stretched state. This work is typically measured in joules (J).

What is work done by a spring?

Work done by a spring refers to the amount of energy released by a spring when it returns to its original state after being stretched. This work is also measured in joules (J).

How is work done to stretch a spring different from work done by a spring?

The main difference between work done to stretch a spring and work done by a spring is the direction of energy flow. When a spring is stretched, energy is input into the system, while when a spring returns to its original state, energy is output from the system.

What factors affect the work done to stretch a spring?

The work done to stretch a spring depends on several factors, including the force applied to stretch the spring, the distance the spring is stretched, and the spring constant, which is a measure of the stiffness of the spring.

Why is understanding work done to stretch a spring important?

Understanding work done to stretch a spring is important in various fields of science and engineering, such as in designing and analyzing mechanical systems, studying the behavior of materials, and understanding the principles of energy conservation.

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