Finding the Transfer Function of a Mass-Spring-Damper System Example

This problem is adapted form a textbook problem that gave me lot of trouble the first time I tried it. It is pretty long, but I think it has a lot of important concepts in it.

Timestamps:
00:00 - Problem introduction and coordinate definitions
01:20 - Free body diagrams and dynamic modeling
13:20 - Laplace Transform
20:07 - Solving system of equations (Cramer's Rule)
40:18 - Finding transfer function
41:55 - Answer

MATLAB code to solve system of equations:
% --------------------------------------------------------------------------------------------------
% Requires Symbolic Toolbox!
syms s F % set up symbolic variables
A = [10*s^2+3*s+2 -2 -3*s; 2 -7-5*s^2 5; 3*s 5 -3*s-5]; % A matrix (coefficients)
b = [F;0;0]; % b matrix (Right hand sides)
X = A\b; % Solve for x vector by multiplying b by the inverse of A
X2 = X(2); % X2 is the second element in our X vector
TF = simplify(X2/F) % transfer funciton is output over input
% --------------------------------------------------------------------------------------------------

If you have any questions or want to see me make a video about something, feel free to leave a comment below!

Errata:
The problem statement has position misspelled as positon

[SaraStigmar]

I learned a lot! The pace is not stressed. You turn over every stone and do it very methodically. Thank you very much for the video! Helpful! :)