A helical compression spring is cut into two halves.What is the resultant stiffness of spring
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Technically, it goes to 0 because you broke it.
If you want to know the spring constant of one of the remaining halves, it’s pretty clear that the spring constant has doubled. To figure this out, all you need to know is the definition of the spring constant: k=δFδxk=δFδx. Let’s start by taking our normal, full-length spring (spring constant k1k1) and stretch it to some length x1x1. We’ll call the total force we’ve applied to the spring F1F1.
Suppose you grab the middle of the spring and hold it perfectly still. Now un-stretch the left half of the spring (the part to the left of where you grabbed it). Nothing about the right half of the spring has changed, so the force being applied to the right half of the spring must (F2F2) be the same F1F1 we found earlier. The difference is that our δxδx has changed. We’re going to call our new δxδx by a new name, x2x2. Now let’s call the spring constant of the right half of our spring k2k2.
So, quick recap. F1F1, k1k1, and x1x1 are properties of the full spring. F2F2, k2k2, and x2x2 are properties of the right half of the spring. By definition of the spring constant, k1=F1x1k1=F1x1 and k2=F2x2k2=F2x2. We’ve shown that F1=F2F1=F2. We grabbed the spring exactly in the middle, so x2=x12x2=x12. Therefore:
k2=F2x2=(F1)(x12)=2F1x1=2k1k2=F2x2=(F1)(x12)=2F1x1=2k1
Therefore k2k2 is double k1k1.
If you want to know the spring constant of one of the remaining halves, it’s pretty clear that the spring constant has doubled. To figure this out, all you need to know is the definition of the spring constant: k=δFδxk=δFδx. Let’s start by taking our normal, full-length spring (spring constant k1k1) and stretch it to some length x1x1. We’ll call the total force we’ve applied to the spring F1F1.
Suppose you grab the middle of the spring and hold it perfectly still. Now un-stretch the left half of the spring (the part to the left of where you grabbed it). Nothing about the right half of the spring has changed, so the force being applied to the right half of the spring must (F2F2) be the same F1F1 we found earlier. The difference is that our δxδx has changed. We’re going to call our new δxδx by a new name, x2x2. Now let’s call the spring constant of the right half of our spring k2k2.
So, quick recap. F1F1, k1k1, and x1x1 are properties of the full spring. F2F2, k2k2, and x2x2 are properties of the right half of the spring. By definition of the spring constant, k1=F1x1k1=F1x1 and k2=F2x2k2=F2x2. We’ve shown that F1=F2F1=F2. We grabbed the spring exactly in the middle, so x2=x12x2=x12. Therefore:
k2=F2x2=(F1)(x12)=2F1x1=2k1k2=F2x2=(F1)(x12)=2F1x1=2k1
Therefore k2k2 is double k1k1.
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