Using first principles, how to get the equation of motion?

[Nicci]

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Show, from the first principles, that the equation of motion of a mass (m) on a spring, subjected to a linear resistance force R, a restoring force S, and a driving force G(t) is given by

d²x/dt²+ 2K(dx/dt) + Ω²x = F(t)

I started by saying S = αx and R = βv.
ma = G(t) - S - R = G(t) - αx - βv where a = (d²x/dt²) and v = (dx/dt)

I am a bit confused on how to use the first principles to derive the equation. Usually for first principles I would have a function like f(x) and then I would use f(x+h), but I am stuck on this part. I did try to use G(t) and G(t+h), but that did not work.
Can someone maybe give me a hint on how to use the first principles in this case? I am sure I will be able to derive the equation if I can just figure out the next step.
Thank you very much in advance.

 

[Andrew Mason]

Show, from the first principles, that the equation of motion of a mass (m) on a spring, subjected to a linear resistance force R, a restoring force S, and a driving force G(t) is given by

d²x/dt²+ 2K(dx/dt) + Ω²x = F(t)

Hi Nicci. Welcome to PF!

Can you provide us with the exact wording of the question? I am not clear on what F(t) is.

AM

 

[Nicci]

Hi Nicci. Welcome to PF!

Can you provide us with the exact wording of the question? I am not clear on what F(t) is.

AM

"Show, from the first principles, that the equation of motion of a mass (m) on a spring, subjected to a linear resistance force R, a restoring force S, and a driving force G(t) is given by

d²x/dt²+ 2K(dx/dt) + Ω²x = F(t)

In your discussion, clearly define S, R, K, Ω, and F(t)."
This is the exact words of the question.

In the textbook it states that:
F(t) = G(t)/m
Ω² = α/m
2K = β/m

Thank you very much.

 

[rude man]

You haven't given any reason for introducing new parameters α and β so why do so?
Other than that, start with ΣF = ma. What is a in therms of x?
How about velocity? In terms of x? And where does G(t) fit in?
So rewrite ΣF in terms of x with the given parameters (typically constants except for G(t).
The rest should flow freely.

 

[Andrew Mason]

"Show, from the first principles, that the equation of motion of a mass (m) on a spring, subjected to a linear resistance force R, a restoring force S, and a driving force G(t) is given by

d²x/dt²+ 2K(dx/dt) + Ω²x = F(t)

In your discussion, clearly define S, R, K, Ω, and F(t)."
This is the exact words of the question.

In the textbook it states that:
F(t) = G(t)/m
Ω² = α/m
2K = β/m

So α is the spring constant (usually denoted as k) which is potentially confusing. Also, F is an acceleration not a force, which can cause confusion.

Do as rude man says and express the sum of all forces in terms of the total mass x acceleration. There are 3 different forces ie:

∑Forces = ma = Applied force + Resistive force + Restorative Spring force

Spring: Let the equilibrium point be 0. How is the restoring force related to x?. Hint: Where is the restoring force always directed toward?

Resistive force: It is not clear from the problem as stated, but this force is a linear function of speed. What is its direction in relation to the speed direction?

Applied force: G(t) is an applied force that is not a function of x.AM

 

[Nicci]

You haven't given any reason for introducing new parameters α and β so why do so?
Other than that, start with ΣF = ma. What is a in therms of x?
How about velocity? In terms of x? And where does G(t) fit in?
So rewrite ΣF in terms of x with the given parameters (typically constants except for G(t).
The rest should flow freely.

Thank you very much. I will define the new parameters and all the other terms.

 

[Nicci]

So α is the spring constant (usually denoted as k) which is potentially confusing. Also, F is an acceleration not a force, which can cause confusion.

Do as rude man says and express the sum of all forces in terms of the total mass x acceleration. There are 3 different forces ie:

∑Forces = ma = Applied force + Resistive force + Restorative Spring force

Spring: Let the equilibrium point be 0. How is the restoring force related to x?. Hint: Where is the restoring force always directed toward?

Resistive force: It is not clear from the problem as stated, but this force is a linear function of speed. What is its direction in relation to the speed direction?

Applied force: G(t) is an applied force that is not a function of x.AM

Thank you very much. It looks like I am on the correct path. I will just define my parameters and other terms.

 

[Wilikka]

Thank you very much. I will define the new parameters and all the other terms.

May you please send me all your solutions 8 am struggling with this question also

 

[kuruman]

May you please send me all your solutions 8 am struggling with this question also

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