Simple Harmonic Motion Springs: work equation

[Addallat]

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Does the negative of potential energy = work done by simple harmonic springs?


I have two different things written down in my notes:

1.

Work = - delta Potential enrgy


so Work = -1/2 (k)x^2


but then i also have written down

2. Since Work = F*d

Work for springs experiencing SHM should be

W = F restoring * delta(x) ~~~~~ > Work = -k(x) * delta (x)



Can someone please let me know what the equation for work is in regards to springs experiencing simple harmonic motion

 

[Captain Sisko]

Does the negative of potential energy = work done by simple harmonic springs?


I have two different things written down in my notes:

1.

Work = - delta Potential enrgy


so Work = -1/2 (k)x^2


but then i also have written down

2. Since Work = F*d

Work for springs experiencing SHM should be

W = F restoring * delta(x) ~~~~~ > Work = -k(x) * delta (x)



Can someone please let me know what the equation for work is in regards to springs experiencing simple harmonic motion

OK. So what you're asking with the bolded has the assumption that force is constant over the distance d, which it is not for the spring. Force changes based on d per the equation:

F = k * x

If you're not comfortable with basic calculus stop reading here.

Energy is really the integral of force over a given distance.

If F = k*x, then Energy is the integral of k*x

If you take the integral of (k*x dx) you get k*x^2/2 | x=0,delta x

Or E = 1/2 * k * (delta x)^2

 

[jtran7]

For a spring think about it in terms of conservation of energy. when you stretch a spring, energy is stored as Elastic potential energy which is equal to 1/2kx^2 this is the total energy that can be converted into kinetic energy. Therefore W which is defined as change in KE is equal to 1/2kx^2.