Problem with proof of elastic energy

[ffrog]

Homework Statement

How do i prove 0.5kx2=elastic energy?

Homework Equations

0.5kx2=elastic energy
Energy in= Energy out

The Attempt at a Solution

I understand how to use the formula but what i don't understand is where the formula comes from. I am guessing it has to do with Fx=Kx but am unsure how K=F then.

 

[Ofey]

The work done can be expressed as

W=Fs

Where F is the force and s is the distance through which the force acts. Since we have a spring we can write this as

W=Fx

Where x is the elongation

Since the force F depends on the elongation (remember F.spring=kx. In other words, when we pull the spring the elongation gets larger and thus we are required to use an even larger force to pull the spring more), the averaga force acting through "x" is

F.average=(0+kx)/2Combining both expressions yields

W=0.5kx^2

 

[ffrog]

thankyou very much problem solved i think i may stick around help afew people out here

 

[LowlyPion]

Homework Statement

How do i prove 0.5kx2=elastic energy?

Homework Equations

0.5kx2=elastic energy
Energy in= Energy out

The Attempt at a Solution

I understand how to use the formula but what i don't understand is where the formula comes from. I am guessing it has to do with Fx=Kx but am unsure how K=F then.

A more precise proof depends on understanding that W = F⋅S as mentioned.

With a spring, however, you have a variable force over the length of its application. So you must use then

PE = W = ∫ F⋅S ds

For F = -kx then

W = ∫ F⋅S ds = ∫ -kx dx = -1/2kx² evaluated from 0 to x

 

[ffrog]

i think i iwill stick with the first one its good enough for me but the secound one makes sense so thanks.