Period of Oscillation in a 1D Linear Spring

[MaxManus]

Homework Statement

The question
I have a one-dimensional linear spring with spring constant E.
The tension is given by σ = Eε, epsilon = strain.. The left side of the spring is held fixed, the right side has a mass m attached to it. We can neglect gravity. What is the natural oscillation period?

What I need help to:
Is it possible to say if it has transverse or longitudinal waves?
Or if I am asking for the wrong hint, please say so.

 

[tiny-tim]

Hi MaxManus!

Is it possible to say if it has transverse or longitudinal waves?

The question says "linear", so you can assume it's longitudinal.

Just do an F = ma equation. ​

 

[MaxManus]

Hi MaxManus! The question says "linear", so you can assume it's longitudinal.

Just do an F = ma equation. ​

Thanks, but I didn't get the F=ma hint.

$$\sigma = E\epsilon$$
for longitudinal

$$\epsilon = \frac{\partial u_x}{\partial x}$$
where u is the displacement field
Longitudinal waves are on the form

u = (u_x,0,0)
u_x = u₀sin(k(x-ct))
where k is the wave number, u₀ is the amplitude and c is the phase velocity.
The equation of motion becomes:

$$\frac{\partial^2 u_x}{\partial t^2} = \frac{\lambda +\mu}{\rho} \frac{\partial^2 u_x}{\partial x^2}$$
Which gives
$$c^2 = \frac{\lambda + 2 \mu}{\rho}$$

And this is as far as I am able to come.

 

[AlephZero]

You are probably confusing yourself by thinking about "waves" at all for this question. It is really about simple harmonic motion of a particle (the mass). Certainly you COULD consider the simple harmonic motion as a standing wave made up of two traveling waves in opposite directions, but that making it a lot more complicated than it needs to be.

Also your use of the word "spring constant" for E seems a bit confused. The equation you give (which is correct) links stress, strain, and Young's Modulus.

What you want is the equation linking force and displacement. That equation involves the length and cross section area of the spring, as well as the value of E.

 

[tiny-tim]

Hi MaxManus!

Thanks, but I didn't get the F=ma hint.

You're analysing the whole spring …

I meant just look at the mass on the end …

d²x/dt² = a = F/m …

carry on from there. ​

 

[MaxManus]

You are probably confusing yourself by thinking about "waves" at all for this question. It is really about simple harmonic motion of a particle (the mass). Certainly you COULD consider the simple harmonic motion as a standing wave made up of two traveling waves in opposite directions, but that making it a lot more complicated than it needs to be.

Also your use of the word "spring constant" for E seems a bit confused. The equation you give (which is correct) links stress, strain, and Young's Modulus.

What you want is the equation linking force and displacement. That equation involves the length and cross section area of the spring, as well as the value of E.

Thanks for the explanation and I now have your equation.
$$\sigma = E\epsilon]$$

$$\sigma = F\A$$
Wikipedia says:
$$\Delta L = \frac{F}{E A} L = \frac{\sigma}{E} L.$$
But not sure what I'm supposed to do with this equation


F=ma
a = d^2 x/\d t^2

 

[tiny-tim]

What force does a spring exert when it is compressed by a displacement x?

 

[MaxManus]

What force does a spring exert when it is compressed by a displacement x?

Spring force:
F = -kx

d2x/dt2 = a = F/m = -kx/n

x(t) = Acos(sqrt(k/m)*t) + Bsin(sqrt(k/m)*t)

But I haven't used the information I was given
F/A = E*epsilon

 

[tiny-tim]

Hi MaxManus!

(just got up :zzz: …)

But I haven't used the information I was given
F/A = E*epsilon

hmm … let's see … the question didn't have a k, instead it had …

I have a one-dimensional linear spring with spring constant E.
The tension is given by σ = Eε, epsilon = strain..

tension (σ) is another name for the force, F

strain (ε) = displacement (x) over original length (L)

so F = Eε = (E/L)x, ie k = E/L

 

[MaxManus]

Thanks again
x(t) = Acos(sqrt(E/(mL))*t) + Bsin(sqrt(E/(mL))*t)

Can I now just do 2pi/T = sqrt(E/(mL))?
T = $$2 \pi \sqrt{\frac{m L}{E}}}$$

 

[tiny-tim]

Hi MaxManus!

(have a pi: π and a square-root: √ )

let's see … period = time to go 2π …

so it's when √(E/mL)t = 2π, ie t = 2π√(mL/E) …

yup! ​

 

[MaxManus]

Thank you so much for all the help and patience.
BTW your smileys really cheer me up