Oscillations of load with spring after rod is suddenly stopped

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The discussion revolves around the dynamics of a load attached to a spring on a rotating rod that suddenly stops. After the rod stops, the second Newton's law is applied to describe the motion of the load, leading to the equation m d²r/dt² = k(r - r₀). The energy from the circular motion is converted into oscillations of the spring, with the oscillation frequency determined by k/m. The participants clarify how to set the initial conditions for the oscillation, particularly the phase angle φ, based on the moment the rod stops. The conversation emphasizes the importance of correctly applying Newton's law and energy conservation to analyze the resulting motion.
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Homework Statement
The rod is rotating with constant angle velocity \omega, load what has weight m may slide on it. The load is held at certain distance by spring with stiffness k and startring length r_0.
Relevant Equations
Find depending r on t if stop the rotating of rod.
I understand that after stopping of rotating I should consider second Newton's law:
m d^2r/dt^2 = k(r-r_0)
And using the law of energy conservation I can propose that energy of circular motion I (\omega)^2/2, where I = mr^2 - moment of intertia will be converted into spring's oscillation. But I not understand what I can do with Newton's law, if there is a constant that disturbs me to consider this motion as oscillations and use the fact that square of oscillation frequency is k/m. Sorry for my bad English, I hope that you can understand this task
 
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alalalash_kachok said:
Homework Statement: The rod is rotating with constant angle velocity \omega, load what has weight m may slide on it. The load is held at certain distance by spring with stiffness k and startring length r_0.
Relevant Equations: Find depending r on t if stop the rotating of rod.

I understand that after stopping of rotating I should consider second Newton's law:
m d^2r/dt^2 = k(r-r_0)
And using the law of energy conservation I can propose that energy of circular motion I (\omega)^2/2, where I = mr^2 - moment of intertia will be converted into spring's oscillation. But I not understand what I can do with Newton's law, if there is a constant that disturbs me to consider this motion as oscillations and use the fact that square of oscillation frequency is k/m. Sorry for my bad English, I hope that you can understand this task
I don’t understand how the rotational energy is relevant. If the rotation of the rod stops then that energy is immediately lost.
Perhaps I have not understood your description. Can you post the original wording in whatever language?
 
Sorry for my misleading, I agree with you. But I have remembered about that I can consider Newton's law for moment after stopping rotating: m $$\frac{d^2r}{dt^2} = k(r-r_0)$$. And try solve this equation, taking $$z = c*e^{i\omega t}$$. After that I get r as superposition of a general solution of the homogeneous equation and the partial non-uniform equation $$r = a \cdot \cos{\omega_0 t + \phi} + r_0$$, where \omega_0 = \sqrt{k/m}. Can I suppose that ##\phi## is 0 just after stopping rotating?
 
alalalash_kachok said:
Sorry for my misleading, I agree with you. But I have remembered about that I can consider Newton's law for moment after stopping rotating: m $$\frac{d^2r}{dt^2} = k(r-r_0)$$. And try solve this equation, taking $$z = c*e^{i\omega t}$$. After that I get r as superposition of a general solution of the homogeneous equation and the partial non-uniform equation $$r = a \cdot \cos{\omega_0 t + \phi} + r_0$$, where \omega_0 = \sqrt{k/m}. Can I suppose that ##\phi## is 0 just after stopping rotating?
You mean $$r = a \cdot \cos(\omega_0 t + \phi) + r_0$$. Curly braces, {}, have special meaning in LaTeX.
The value of ##\phi## depends on how you define t=0. If that is the instant the rod stops rotating then you need a value of ##\phi## for which ##t=0 ## gives ##r=r_0##.
 
haruspex said:
You mean $$r = a \cdot \cos(\omega_0 t + \phi) + r_0$$. Curly braces, {}, have special meaning in LaTeX.
The value of ##\phi## depends on how you define t=0. If that is the instant the rod stops rotating then you need a value of ##\phi## for which ##t=0 ## gives ##r=r_0##.
Okay, assume that t=0 is moment when the rotating rod is stopped. After that the load starts spring fluctuations, don't you think? Can I consider Newton's law: $$m (\omega)^2 r_1= k(r_1-r_0)$$ for the moment just before stopping and take from this starting value of r(t)? Then put it in equation r(t) as solution of the partial non-uniform equation
 
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alalalash_kachok said:
Okay, assume that t=0 is moment when the rotating rod is stopped. After that the load starts spring fluctuations, don't you think? Can I consider Newton's law: $$m (\omega)^2 r_1= k(r_1-r_0)$$ for the moment just before stopping and take from this starting value of r(t)? Then put it in equation r(t) as solution of the partial non-uniform equation
Sounds right. Please post what you get.
 
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