Understanding this Equation of motion with a constant

[Jamie2020]

Homework Statement

equation of motion

Relevant Equations

x = b log (at)

x = a t exp(-b t)

x = exp(-b t) sin(a t)

x = a sin(t)/b

Completely new to this and wondered if someone can explain what the correct equation of motion is if x is extension, t is time and a,b are constants

x = b log (at)
x = a t exp(-b t)
x = exp(-b t) sin(a t)
x = a sin(t)/b

 

[etotheipi]

Hi Jamie!


What is the motion? Do you have a diagram or something that you might be able to upload?

 

[Jamie2020]

Hi, sorry i forgot to include the main text:

Some weight is added to a spring. The spring is then extended and released. The extension of the spring oscillates with decreasing amplitude.

Hope this helps?

 

[etotheipi]

If you were to try sketching a graph of displacement ($x$) against time for this damped oscillator, what might it look like? How might you go about writing an equation for it?

 

[cnh1995]

Hi, sorry i forgot to include the main text:

Some weight is added to a spring. The spring is then extended and released. The extension of the spring oscillates with decreasing amplitude.

Hope this helps?

The spring "oscillates". Does that eliminate any of the given options?

 

[Jamie2020]

I think the answer is x = a sin(t)/b

 

[etotheipi]

I think the answer is x = a sin(t)/b

What did your graph look like?

 

[Jamie2020]

The red line is what was already on the sheet and the blue line is what i believe the graph would look like?
Thanks

 

[etotheipi]

View attachment 261777

The red line is what was already on the sheet and the blue line is what i believe the graph would look like?
Thanks

Hmm, not quite. Let's forget about the business with the reducing amplitude for now. Just focus on the oscillatory motion with no damping.

If you have a vertical spring with the mass attached to the bottom, and you give the mass a little push downward, it starts to vibrate up and down, right? Say you recorded how high it was every 0.001 seconds (you're very patient... and fast!). You then plot the height of the mass against time on your graph. What would a graph of it's height against time look like (take $h=0$ to be where you release the spring from, just for simplicity).

 

[Jamie2020]

Hmm may have confused myself but would it look like

 

[etotheipi]

Hmm may have confused myself but would it look like

Not really. If there's no damping then the thing's just going to keep oscillating up and down forever. Up and down... rinse and repeat... try getting a large rubber band and hanging a mass on that, and giving it an initial push. What motion results?

 

[Jamie2020]

How about, something like

 

[kuruman]

Close enough. Which of the given choices most likely would look like this when plotted?

 

[etotheipi]

That's more like it! Sort of.

For undamped SHM, you'll get a perfect sine/cosine:

Note that the $x$ axis is still the centre of the oscillation, there's no migration of the centre of oscillation downward as you have drawn.

You've sort of jumped the gun and have started reducing the amplitude. If you do damp the motion you end up with a graph like this:

which is similar to what you drew. Now, you're in a position to answer the question. What two functions does this second graph remind you of?

 

[Jamie2020]

I'm now going for

x = exp(-b t) sin(a t)

??

 

[etotheipi]

Yeah, that's the one.

The $e^{-bt}$ bit is called an envelope function. In this case, it's always less than 1 for $t> 0$. You can imagine first drawing the sine wave, and then going along to every point and scaling the displacement by a factor of $e^{-bt}$. This factor gets smaller as time progresses, and the amplitude of oscillation just gets smaller and smaller. Makes sense intuitively, right?

 

[Jamie2020]

Yes I get some understanding on this now. Thanks for your help. Side question, I don't suppose you know what each letter in the answer above represents on a graph so I know for next time and can visualise what is going on?

Thanks again

 

[etotheipi]

I'd recommend going to https://www.desmos.com/calculator and punching this exact text into the box:

y=Ae^{-bx}\sin\left(ax\right)

Click the blue "all" button and you'll get a bunch of sliders. Try varying them and see what happens! I'd recommend making little $a$ about 5, little $b$ about 0.4 and big $A$ about 4 as a start.

As for a more mathematical answer, in physics at least the equation for damped SHM is usually written in the form $x = Ae^{-\gamma t}\cos{\omega t}$. $\gamma$ is called the damping coefficient (bigger value means more damping!) and $\omega$ is the angular frequency, $\omega = \frac{2\pi}{T}$.

You can derive all of these equations from scratch if you setup a differential equation, which might be a good exercise to try sometime! If you don't have external forcing, it's a homogenous second order ODE.

 

[kuruman]

Yes I get some understanding on this now. Thanks for your help. Side question, I don't suppose you know what each letter in the answer above represents on a graph so I know for next time and can visualise what is going on?

Thanks again

The first choice represents nothing physical; what happens at t = 0?
The second choice is a special case of a critically damped harmonic oscillator.
The third choice represents an underdamped harmonic oscillator.
The fourth choice represents harmonic motion without damping.