# Total Energy Of A Spring-Mass System

Hello everyone, I hope someone can help me with this simple harmonic motion question
I've been trying it for about an hour now and I think i may have found the problem but I want to run it by someone else :)
I'm taking this Phsyics 12U course (Uni prep for Ontario, Canada) through correspondence and all I have to go by are the work books that I have received.

## Homework Statement

My work book gives me a bunch of formulas for harmonic motion, those formulas are what I am using to re-arrange and sub and so on.

Prove that Max Speed of a mass on a spring is given by 2 * pi * f * A

So after re-arranging and all that fun stuff I get as far as

vmax= sqrt(k/m) * A

===============================================

so now the speed of the mass will be at max when it is right at equilibrium (x = 0)
and the potential energy will be zero and the kinetic energy will be max

So then I get this formula using energy equations, (Ek = kinetic energy, m = mass)

vmax = sqrt(2 * Ek / m)

now we can get Ek another way, and that is using the total energy of the system equation and this is where i get hung up

in my workbook, it has the total energy of the system as

Et = kA^2

the total energy is all kinetic in this case so we can sub Ek for Et

so I try to plug that into

vmax = sqrt(2 * Ek / m)

and I end up with

sqrt(2k/m) * A

which kind of looks like the 2 * pi * f * A rearranged except for that useless 2 in the numerator! :@

so then I research this using google and on wikipedia it has the total energy of a spring mass system as

Et = (kA^2) / 2

so this formula has the same thing except the 2 in the denominator
I then check another site and it also has the 2 in the denominator

and it clicks, i put this formula in the equation instead and it all works out!

so is my workbook giving me bad information??

is the total energy of a Spring Mass system given as

Et = (k * A^2)

or

Et = (K * A^2) / 2

???

thanks in advance for all of the help!

Last edited:

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alphysicist
Homework Helper
Hi D3SI,

Yes, you found the problem; there should be a (1/2) factor in the total energy formula.

woot!