Total Energy Of A Spring-Mass System

In summary, the conversation discusses a question about simple harmonic motion and the formula for maximum speed of a mass on a spring. The person is taking a physics course through correspondence and is using their workbooks as a reference. After some rearranging and researching, they discover that the total energy formula given in their workbook is missing a (1/2) factor, leading to incorrect results. They plan to address this issue with their school.
  • #1
D3SI
2
0
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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  • #2
Hi D3SI,

Yes, you found the problem; there should be a (1/2) factor in the total energy formula.
 
  • #3
woot!

thanks for the reply :)

gunna have a talk with the adult ed people at the high school, looks like they don't check the books b4 using em :(
 

1. What is the equation for the total energy of a spring-mass system?

The total energy of a spring-mass system is given by the equation E = 1/2kx^2 + 1/2mv^2, where k is the spring constant, x is the displacement from equilibrium, and m is the mass of the object.

2. How does the total energy of a spring-mass system change with displacement?

The total energy of a spring-mass system is directly proportional to the square of the displacement from equilibrium. This means that as the displacement increases, so does the total energy of the system.

3. Does the total energy of a spring-mass system change with mass?

Yes, the total energy of a spring-mass system is directly proportional to the mass of the object. This means that as the mass increases, so does the total energy of the system.

4. What happens to the total energy of a spring-mass system when the spring constant increases?

If the spring constant increases, the total energy of the system increases as well. This is because a higher spring constant means that the spring is stiffer, and therefore more energy is required to stretch or compress it.

5. Can the total energy of a spring-mass system ever be negative?

No, the total energy of a spring-mass system cannot be negative. This is because the equation for total energy includes only positive terms, and energy is always conserved in a closed system.

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