00:01
So in this example, we're going to figure out how to derive our equations of motion for constant angular acceleration.
00:09
That is, this alpha here is a constant.
00:14
And we're going to look at these three equations in particular, starting with the first one here.
00:19
So for the first one, we just need to recall that the definition for angular acceleration alpha, when it doesn't change, is just delta omega, change in angular velocity velocity.
00:33
Over the change in time delta t.
00:36
In the exact same way that for linear acceleration, it would be defined as delta v over delta t for a constant linear acceleration.
00:48
So if we take this equation here and we solve for delta omega, by multiplying both sides by delta t, we will end up with delta omega equals alpha delta t.
01:02
And if we expand delta omega, knowing that it's defined as omega final minus omega initial equals l delta t, we get our first equation there.
01:22
Not so bad at all.
01:24
All right.
01:26
So now we're going to look at the second equation here, and this one's going to require a little bit more work.
01:31
The way i think this one's easiest to figure out is by considering the plot of omega against time.
01:40
And the reason i say this is because if you remember during your study of linear motion, we could determine a displacement, a linear displacement by looking at a plot of linear velocity versus time.
02:00
And in some interval from t0 to t1, finding the area under the curve.
02:14
And this area would correspond to a linear displacement from t0 to t1.
02:23
Similarly, we can do the exact same thing for angular displacements.
02:31
So in the exact same sense, if i were to take some interval here and find the area under this curve, omega versus t, this would correspond to the angular displacement, delta omega.
02:48
So let's try to come up with a plot of this equation here and what it would look like in general.
03:03
So if we take omega initial to be a constant, we solve for omega final here.
03:15
So omega final is omega initial plus alpha times delta t.
03:21
We've already decided that alpha is a constant.
03:25
And we're going to be plotting this graph with omega final in the y -axis.
03:32
And delta t on the x -axis.
03:39
So what this is going to look like, if we remember how to plot a line here, we can see that omega initial is going to be our, it's going to be the point that the line intersects the vertical axis here.
03:55
So if we think back to like y equals mx plus b, right? this is m and this is x and this would be our b.
04:06
So right here is going to be the point that the line intersects the y -axis, and this has a height of omega initial.
04:17
And our line might look something like this, and its slope is going to be alpha.
04:26
And then this is our independent variable here, delta t.
04:31
So the area under this curve from zero all the way out to, you know, our final delta t, maybe something like here, is, or the easy way to calculate it would be to break it up into two separate shapes here, where we have one box shape on the bottom and then a triangle up top.
04:57
So at the end of this interval, we know that we'll be ending at omega final.
05:06
The width of this whole thing is just delta t, right? we go from delta t equals zero to delta t.
05:14
The height of this here is just omega initial.
05:17
And the height of this here is omega final minus omega initial.
05:27
So this one's omega initial...