Energy stored in a spring

[pigeon_soup]

I am helping a friend with his coursework from college and we have done all but the final question of the distinction,in the second part of a question after finding the expansion of a brass rod from an increase temperature he has to calculate the force stored in the spring after it has been compressed, the spring has a stiffness (k) of 20,000 kN/m and has been compressed 0.0756mm. Having looked through several books and on the interwebs we have found this formula:

 

strain energy on the spring under tension or compression= ½kx²

 

where k is stiffness and x is compression

 

we believe this is the correct formula but want to check just in case.

 

if you have any corrections or could comfirm this is the correct formula it whould be much apreciated :confused:

 

pigeon

[CraigD]

strain energy on the spring under tension or compression= ½kx²

 

where k is stiffness and x is compression

Provided that the spring’s stiffness is uniform – that is, that for all compression distances x the force necessary to compress it by 2x is 2 times that to compress it by x – this formula’s OK.

 

If, like me, you’d prefer deriving it to searching it up in textbooks or the www (I find it easier to remember a few things about mechanics, algebra, and calculus, then derive what I need when I need it), it goes like this:

 

The definition of work and energy is: [math]W = E = F \Delta x[/math], where [math]F[/math] is force, and [math]\Delta x[/math] is change in position.

Stiffness is defined as: [math]k = \frac{F}{\Delta x}[/math], where [math]F[/math] is the force required to compress a body by distance [math]\Delta x[/math].

So, via Eulers’ rule, [math]E = \int k x \, dx = \frac12 k x^2[/math].