Undergrad Visualizing the Fourier transform using the center of mass concept

[person_random_normal]

I found this video on youtube which is trying to explain Fourier transform using the center of mass concept


At 15:20 the expression of the x coordinate is given in the video. I believe it is wrong, and it should be:

$\frac{{\int g(t)e^{(-2 \pi ift)}.g(t).2 \pi f.dt}} { \int g(t).2 \pi f.dt}$

Because the wire is assumed to have- uniform mass distribution
Can someone please check?

 

[Andrew Mason]

At 15:20 the expression of the x coordinate is given in the video. I believe it is wrong, and it should be:

$\frac{{\int g(t)e^{(-2 \pi ift)}.g(t).2 \pi f.dt}} { \int g(t).2 \pi f.dt}$

Because the wire is assumed to have- uniform mass distribution
Can someone please check?

The math in the video is correct. He doesn't really explain the centre of mass concept (he is giving all sample points the same mass) and he leaves out some steps. He goes from:

$$\hat{x}_{com} = \frac{1}{N} \sum_{k=1}^N g(f)e^{-2\pi ift}$$

directly to:

$$\hat{x}_{com} = \frac{1}{(T_2 - T_1)}\int_{T_1}^{T_2} g(f)e^{-2\pi ift}dt$$

without explanation.

The intermediate steps should be something like:

$$\hat{x}_{com} = \frac{\Delta t}{(T_2-T_1)}\sum_{k=1}^N g(f)e^{-2\pi ift}$$

where $\Delta t = (T_2-T_1)/N$ i.e. $\Delta t$ is the time interval between equally spaced sample points so: $N = (T_2-T_1)/\Delta t$

It follows that:

$$\hat{x}_{com} = \frac{1}{(T_2-T_1)}\sum_{k=1}^N g(f)e^{-2\pi ift}\Delta t$$

and in the limit where $\Delta t \rightarrow 0$:

$$\hat{x}_{com} = \frac{1}{(T_2 - T_1)}\int_{T_1}^{T_2} g(f)e^{-2\pi ift}dt$$
AM

 

[SmolBoi96]

I am confused about this as well. Here is my take on the problem. I think instead of center of mass what Grant defined is actually the center of time or the time average. Let's compare the limiting process for the center of mass and the time average to go from the discrete case to the continuous case.

For the center of mass we have:

In the limiting process we take smaller and smaller mass pieces. This will lead to the integral you derived with a g square integral on the top and a g integral at the denominator.

What Grant did was to keep the mass of each piece constant and take more and more pieces at smaller time intervals.

This limiting process leads to finding the time average of the position of the winding curve in 2D. Notice that compared to the center of mass integral dm is just replaced with dt. The process of finding the center of mass for more and more particles of constant mass leads to the time average. This will lead to the g integral on top and a constant integral at the bottom.