This is a problem from a systems dynamics textbook the solution is here https://

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This is a problem from a systems dynamics textbook the solution is here https://www.chegg.com/homework-help/System-Dynamics-3rd-edition-chapter-4-problem-19P-solution-9780073398068

I'm working to understand the solution better. From what I can tell, it seems that the method involves first drawing a FBD and writing equations for the net forces.
The small pulley with the spring has the equation of 2T=1/2kx
The mass block has the equation of T=f+mg (assuming equilibrium).
Newton's second law relates force to motion, so assuming there is motion since the problem asks for the equation of motion, the motion of the block is described by F=ma. The FBD gave us the equation for the net forces, T=f+mg, moving T to the other side gives the expression, f-T+mg. Setting this equal to ma or mx'' (x''=xdouble-dot).
mx''=f-T+mg...
I see in the solution they replace T by using the force equation created from the FBD with the pulley since they are coupled which I am cool with. So 2T=1/2kx means T=1/4kx, replacing T in the mx'' equation with 1/4kx
mx''=f-1/4kx+mg...so this is where my question begins. My initial question is why did we replace T with 1/4kx instead of just keeping T in the equation?
But, once I see that T can be expressed in terms of x, then I kinda-sorta see why that T term (1/4kx) then get's moved to the side with the x'' since they both have x in them, but I don't fully understand why this is. Also, If I hadn't seen the solution I wouldn't have thought that I should even replace the T term. I am confused enough that even trying to frame a question to address my confusion is difficult but this is my attempt at trying to understand how you know to replace T in the first place and how it conveniently could be expressed in terms of x and what this x term means in terms of Newton's second law and why it get's moved to the side with the x''.

Also, I did find this passage in the textbook on page 128
"In Example 3.2.3, it is inconvenient to use an energy-based analysis to compute x(t) or the tensions in the cable. To do this it is easier to use Newton’s law directly."

This kind of addresses not using T, but honestly, it doesn't clearly explain why it's inconvenient.

Figure P4.19 4.19 In the pulley system shown in Figure P4. 19, the input is the applied force f, and the output is the displacement x. Assume the pulley masses are negligible and derive the equation of motion

Explanation / Answer

I understood your question clearly that why did we replace T with 1/4kx instead of just keeping T in the equation?

Because our mass equation already in terms of x which is mx''=f-T+mg, and this is the second order differential equation if you are not replacing T with 1/4kx then for the solution of this differential equation you need the value of T, which is generally not given.This is just a substitution of T into 2nd equation. If you have the value of T then you don't have to replace T in terms of x. And by keeping T in the equation you can find only displacement of the solution. You can't find system's natural frequency which is the square root of the ratio of coefficient of x to the coefficient of x''

NOW, it is inconvenient to use an energy-based analysis to compute x(t) or the tensions in the cable. To do this it is easier to use Newton’s law directly." It means that, if you are using energy method then you will have to find energy of each component in the system, it is sometimes seems to be difficult to finding energy of each component. Where Newton's law only need free body diagram to solve any problem.

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