00:01
Simple harmonic motion appears in a lot of places.
00:04
And here we're going to see that it appears in a piston sitting atop an ideal gas, such as air or nitrogen.
00:16
And so what we'll be needing to look at this problem is we will, of course, need newton's second law, which says that the sum of forces on that piston is equal to its mass times.
00:33
Acceleration.
00:36
And here we're really just considering the by direction is where things will get out of place in terms of equilibrium.
00:46
We'll also need the relationship between force and pressure.
00:51
So force per unit area is equal to pressure.
00:59
And finally, we will need the ideal gas law with the assumption that it is an ideal gas.
01:09
Under the piston.
01:13
Pressure times volume is number of moles, times gas constant, times temperature.
01:21
Okay, now we are given some parameters in our system, the mass of the piston, its radius, the height it's sitting at, and the pressure outside, and we are going to be displacing from equilibrium.
01:39
So this is the equilibrium situation.
01:42
And in order to work this out, we're going to start off with the piston in equilibrium.
01:53
So what we know there is the sum of the forces in the y direction is zero.
02:00
And let's draw what those forces look like.
02:04
There will be a force from the ideal gas upwards.
02:10
I want to call that.
02:15
That's from the pressure and the gas at equilibrium.
02:18
So just to point out that that's coming from the ideal gas.
02:25
And there will be a force downwards, a couple forces downwards, one from the outside air.
02:38
Okay, so we're just turning pressures into forces, multiplying by the area of the piston, assuming it is circular, pi r squared.
02:50
And then there's also the weight.
02:53
Yes, there's always the weight, m .g.
02:59
And we know those all have to sum up to give us zero.
03:04
So the sum of those forces, the ups have to equal the downs.
03:10
So the pressure at equilibrium times the area has got to equal the p.
03:18
Not outside pressure times area plus m g.
03:27
Okay, and i rather suspect, of course, we'll draw a new diagram with it out of equilibrium and find the net force there, which is going to be related to the acceleration, and hence the simple harmonic motion.
03:43
But i rather suspect we're going to need the equilibrium pressure, and i'll just solve for it.
03:51
It's going to look a little messy, so this is probably the only place.
03:56
We'll write this down, and we'll just use a substitution later.
04:02
With all these symbols.
04:05
But just to prove that we know what this quantity is, if we know the pressure outside, plus the mass of the piston, and the area, the radius of the piston.
04:22
So these are all constants of given in the problem, supposedly.
04:28
We won't worry about what they particularly are.
04:34
Okay, now for the interesting part, we're going to displace the piston by a small y, much less than h in the upwards direction.
04:49
And then we have the sum of the y forces is equal to m times the acceleration.
05:00
And we can draw our free body diagram.
05:02
It's not going to look too different.
05:05
It's going to look pretty identical, except we're going to have a different pressure upwards, which we are going to have to solve for and call that pressure for d for displaced.
05:31
And that's what we're going to have to figure out in order to write our balance because the other quantities.
05:44
Are the same.
05:48
Oops, let's try to show them the same anyway.
05:52
Let's see.
05:54
I originally drew m .g a little bit bigger than the pressure from upstairs.
06:06
Ok, but it's important to note that these are the same.
06:09
The pressure from the ideal gas will be different.
06:17
And i'll just show again that that's coming from the ideal gas, because we're going to solve for the pd by using the ideal gas law.
06:39
And then i put a question mark there just and then put it into newton 2nd.
06:58
Okay, so ideal gas law.
07:03
I'll just write it down that we had the pressure at equilibrium times the volume at equilibrium...