00:02
Once again, welcome to new problem.
00:06
Still dealing with linear densities.
00:09
Remember linear density, when you think about it, it's in terms of grams per meter, for example.
00:19
So if you say, for example, you have two strings, you have two strings like that, and there's a connection.
00:34
Right here.
00:35
There's another one.
00:38
And this one is the first string.
00:41
We're going to call it s1.
00:43
Remember s1 is string number one and then we also have s2 which is string number two.
00:55
So we have two strings.
00:58
The gap between them is 4 .0 meters.
01:10
So that's the the entire breadth of the string and this is the connection.
01:19
The first string has a length of l1 and the second string has a length of l2.
01:30
So those are the two strings we're looking at.
01:34
And there's a connection in the middle.
01:36
There's a connection in the middle.
01:39
In terms of linear density, in terms of linear density, we have 2 .0 grams per meter.
01:48
That's the linear density of the stream.
01:52
2 .0 grams per meter.
01:59
And that's the linear density of the fast string.
02:04
And then for the second string, the linear density happens to be 4 .0 grams per meter.
02:15
That's the linear density for the second one.
02:20
So if we do have connections right here, if we do have connections right here, and think about the strings and you have pulses going in one direction and then pulses going in the opposite direction.
02:42
So, you know, we have pulses in both directions.
02:48
So we have someone sending pulses in both directions.
03:00
And the way that happens is, remember, we have a connection right here in the middle.
03:06
So using this connection, we're going to have pulses going that way, and we're going to have pulses going that way in both directions.
03:20
So assume that these two pulses, the pulses get to the string ends at the same time.
03:42
So in terms of time factor, the time it takes for a pulse to move from right here in the middle along the first string to the end, and then from right here in the middle, along the second string to the end, is exactly the same.
03:59
So they have the same time factor.
04:01
The question is, what's the length of string one and what's the length of string two? the relation that we're going to pick up is based on the speed of the wave.
04:23
That's one thing we're going to look at in solving the problem.
04:26
We're also going to look at the tension in the string.
04:31
And then don't forget the linear mass density, the linear mass density.
04:43
You know, we can add that.
04:45
These were linear mass densities that were given in the problem.
04:53
So the speed, the speed is dependent on the tension and the linear mass density.
05:01
These two are influencing factors when it comes to the speed.
05:05
Of the wave itself.
05:08
For string one, for string one, velocity one is radical the tension in string one of the linear mass density in string one.
05:24
This is the tension in string one, and this is the tension in string one, and this is the linear mass density in string one.
05:42
This is the velocity.
05:45
This is the velocity in stream 1.
05:50
That's the velocity.
05:52
So we've identified the tension.
05:55
And so we're going to go ahead and simplify this.
05:59
If v1 equals to radical ts1 of a new one, we want to get rid of the radical, so we square both sides of the equation.
06:10
So of v1 equals to t of v1 squared equals to tension in spring 1 over mi 1, multiplying both sides by the linear mass density, these two cancel.
06:26
So the tension in string 1 is the velocity in string 1 squared and the linear mass density for string 2.
06:37
For the second string for the second string string 2 we do have a v2 equals to radical t s2 of the mu 2 on a square both size of that equation so this becomes tension in string 2 of a mu 2 and whether mu is the linear mass density and then with you know, with that type of understanding, we can go ahead and see that the velocity, the same velocity, the general velocity, is distance over time.
07:50
That's our velocity.
07:51
Our velocity is distance over time.
07:54
We can call it d over t.
07:56
And so this is the same as radical t over mew.
08:01
In general, to get the distance, we multiply both sides by t, so the distance traveled is vertical, the tension of a mu, times the time itself.
08:14
Maybe we can shift the time here for purposes of convenience, right here, this is t.
08:24
The times are the same for both waves, so string, we have stream one and string two.
08:31
In terms of time, they're exactly the same, t1, t2, so t1 equals to t2.
08:43
In terms of tension, we have tension in 1 and tension in 2, tension in string 1 and tension in string 2.
08:55
These two tensions are also the same.
08:59
The two tensions are also the same...