Tip:
Highlight text to annotate it
X
Well, normally I would solve this algebraically and I think that's the best way to do it
and I encourage you to do it in the forums.
But since we already calculated Vmax numerically using the numbers
I had made up last time, k = 500, m = 1, and Xmax = 0.2.
Let's go ahead and get ourselves an answer. So how do we do this?
Well, I know that the system has a total amount of energy.
When it is at the extreme points, the energy is all potential.
In the equilibrium, it's all kinetic and this middle point here is both kinetic and potential.
I can phrase this mathematically as the total energy is going to be
the kinetic plus the potential.
Well the total energy is just whatever energy I had when I pulled it all the way back to Xmax.
The kinetic energy? Well, I'm not sure what that is. But I know that this is the equation that describes it.
And actually, I know all of these numbers except for v.
To solve these for v is just a matter of doing a little bit of massaging, which I will leave to you.
When I carry out all these algebra and arithmetic, I find that when I'm at this point
when the mass is moving through this position here, the velocity is 0.87.
It's maximum velocity. 87% and it's moving pretty fast.
My plot will go up to here, up here, and probably somewhere around here.
Now, I could do very similar algebra to find that there's also a negative solution
and this solution corresponds to the exact same thing but when the mass is moving this way.
I could also find that there are solutions over here and if I solve for the position
when the velocity was 1/2 of its maximum I would get 0.87 Xmax.
I could put points here and here.
Can you see what shape is starting to emerge on the points we're plotting?
It's a circle or at least it's supposed to be. Sorry some of my points are a little bit off.
This is something in Physics that we call a phase diagram.
And it's a very powerful tool once you understand how to use it.
The way to interpret this diagram is as follows.
We start with a mass put all the way back to Xmax. We let it go.
Well in this case, it starts with negative velocity. It moves that way.
That would correspond to going into the negative realms of velocity down here.
Eventually, it maxes out in speed as it passes through equilibrium
but keeps moving to the left as it can be seen by the negative velocity.
Eventually, it reaches this extreme over here and finally it turns around.
Positive velocity moving to the right passes through the equilibrium at maximum speed
and keeps moving through the right and returns to where it started.
The Wikipedia page in simple harmonic motion has a really great illustration.
Now, a couple of things that I notice that are different about this illustration is
position is vertical, velocity is horizontal, the spring itself is moving up and down
but you get the same idea.
Here, the vertical position is tracking the object's position
and as the object moves downwards we see that the ball is on the left side of this line
corresponding to a negative velocity as it moves upwards.
This ball on the right side is corresponding to a positive velocity.
Okay, so why did I spend all this time to show you this?