00:01
Let's start with the radial vector.
00:05
It's going to be equal to 3t, t squared, and t cubed.
00:10
Now in order to find the curvature, without going into what it is geometrically, the curvature is going to be equal to the magnitude of the cross product of the velocity with the acceleration divided by the magnitude of the velocity cubed.
00:33
We're not going to prove why this is the case, but we'll just utilize this to find the curvature.
00:39
We have r, so let's find the derivative.
00:41
R prime of t will be equal to, taking the derivative of each component, 3t, 3t squared.
00:52
And r double prime of t will be equal to 0 .260.
01:02
Now the magnitude, we're not too worried about the r double prime, but we want to know the magnitude of r prime of t.
01:09
That will be baggerin's theorem.
01:11
So we'll take the square root of 3 squared plus t squared plus 3t squared squared.
01:22
So that will give us 9 plus 4t squared plus 9t to the fourth power, all square rooted.
01:37
So that's the magnitude.
01:40
We have the vector for the velocity and acceleration.
01:43
Now let's take the cross product.
01:44
So r prime, for us, r double prime will be equal to the determinant of the matrix.
01:59
We'll put r prime first.
02:01
So r prime was 3t, 3t squared.
02:06
And r double prime was 0, 2 and 16.
02:12
So if i expand this out, i'll get 5 .12t squared minus 6t squared.
02:29
And then we'll get 18t minus 0.
02:39
And then in the k hat, we will have 3 times 2 is 6 minus 0.
02:46
So the vector is 6t squared, negative 18t and 6.
02:57
So the magnitude of this vector, the magnitude of r prime cross r double prime will be equal to the square root of each component.
03:10
So that will be 36t to the fourth plus 18 squared, which is 324t squared plus 36.
03:29
So the curvature will be equal to the magnitude of r prime of t cross r double prime of t over the magnitude of r prime of t cubed.
03:50
And that will be 36t to the fourth plus 324t squared plus 36.
04:03
That's the magnitude part.
04:05
I know it's under a square root, but so is the r cubed.
04:09
So yeah, let me actually write this as a square root...