Are the Force Equations for Rotational Motion Accurate?

AI Thread Summary
The discussion focuses on the accuracy of force equations for rotational motion, specifically addressing the equations for two blocks, m and M. The original poster expresses confusion about their equations and suggests that using energy methods might be more effective. Participants discuss how to express potential energy U(r) in terms of r and the implications of differentiating the energy equation. Ultimately, the conversation highlights the importance of understanding the relationship between angular momentum, energy, and the dynamics of the system, leading to a more elegant solution. The exchange emphasizes the learning process and the value of problem-solving in physics.
Bling Fizikst
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Homework Statement
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Relevant Equations
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Screenshot 2024-03-24 212556.png


Writing force equations for block ##m## : $$T-mr\omega^2=m\ddot{r}$$ and for block ##M## : $$Mg-T=M\ddot{r}$$ I think there are mistakes in my equations as they are leading to nowhere and morever i think force methods are really risky in this regard . It would be better to write the total energy of the system which i don't know how to . I tried to write $$\frac{1}{2}m\dot{r}^2 +\frac{1}{2}M\dot{r}^2 + \frac{L^2}{2mr^2} +U(r)=E$$ where ##L=mR^2\omega##
 
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Have you done any Lagrangian mechanics?
 
PeroK said:
Have you done any Lagrangian mechanics?
No
 
Bling Fizikst said:
No
It should come out from conservation of AM and energy. You're aiming for an equation for ##\ddot r##. Which you should be able to get from:

Bling Fizikst said:
$$\frac{1}{2}m\dot{r}^2 +\frac{1}{2}M\dot{r}^2 + \frac{L^2}{2mr^2} +U(r)=E$$ where ##L=mR^2\omega##
 
... differentiating the energy equation should work.
 
PeroK said:
... differentiating the energy equation should work.
I did that but how do i deal with the potential energy of the system : ##U(r)## $$(m+M)\ddot{r}-\frac{L^2}{mr^3}+U'(r)=0$$
 
Bling Fizikst said:
I did that but how do i deal with the potential energy of the system : ##U(r)##
You should be able to express that in terms of ##r##.
 
PeroK said:
You should be able to express that in terms of ##r##.
Need help on this , really deadstuck . Writing forces and using ##\frac{-\partial U}{\partial r}=F## doesn't seem to help?
 
Bling Fizikst said:
I did that but how do i deal with the potential energy of the system : ##U(r)## $$(m+M)\ddot{r}-\frac{L^2}{mr^3}+U'(r)=0$$
What precisely is stopping you from writing down an expression for ##U(r)##?
 
  • #10
Bling Fizikst said:
Need help on this , really deadstuck . Writing forces and using ##\frac{-\partial U}{\partial r}=F## doesn't seem to help?
How does the position of the mass M depend on ##r##? Take ##r = R## as the zero point.
 
  • #11
Is it -Mgr:nb)
 
  • #12
Bling Fizikst said:
Is it -Mgr:nb)
What happens to ##M## as ##r## increases?
 
  • #13
it goes up so +Mgr? please forgive me if i am acting dumb , this is my first time using the potential energy function in this way . It gives me the wrong ans anyways , so it is wrong
 
  • #14
Bling Fizikst said:
it goes up so +Mgr? please forgive me if i am acting dumb , this is my first time using the potential energy function in this way
Yes. Note that ##Mg## is related to ##L## by the initial equilibrium.
 
  • #15
$$(m+M)\ddot{r} -mr\omega^2- 2mr\omega^2=0$$ $$\implies \omega'=\sqrt{\frac{3m\omega^2}{m+M}}$$
 
  • #16
Bling Fizikst said:
$$(m+M)\ddot{r} -mr\omega^2- 2mr\omega^2=0$$ $$\implies \omega'=\sqrt{\frac{3m\omega^2}{m+M}}$$
That's the right answer, but I'm not sure quite how you got there.
 
  • #17
PeroK said:
That's the right answer, but I'm not sure quite how you got there.
##F = -kr## implies ##\omega = \sqrt{k/m}##.
 
  • #18
vela said:
##F = -kr## implies ##\omega = \sqrt{k/m}##.
No need to consider only small oscillations?
 
  • #19
Oh, you were asking about the steps preceding that last step. Yeah, the OP isn't exactly forthcoming with his reasoning.
 
  • #20
Bling Fizikst said:
I did that but how do i deal with the potential energy of the system : ##U(r)## $$(m+M)\ddot{r}-\frac{L^2}{mr^3}+U'(r)=0$$
I wonder whether you replaced ##L^2## with ##mr^2w## here? To get from the above equation to this:
Bling Fizikst said:
$$(m+M)\ddot{r} -mr\omega^2- 2mr\omega^2=0$$
In any case, the solution only applies to small oscillations. You will need an approximation somewhere.
 
  • #21
PeroK said:
I wonder whether you replaced ##L^2## with ##mr^2w## here? To get from the above equation to this:
Yes
 
  • #22
Note that ##w## is a constant, like ##M, g## and ##R## and is the fixed angular velocity at the equilibrium. ##r## and ##\dot \theta## are the variables. We have:
$$L = mR^2w = mr^2\dot \theta$$Also, with ##U(r) = Mgr##, we have:
$$\frac d{dt} U(r) = Mg\dot r$$And
$$\frac d {dt}\dot r^2 = 2\dot r \ddot r$$
 
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  • #23
Anyways , i found a really elegant solution in David Morin :
1711305500535.png
 
  • #24
Bling Fizikst said:
Anyways , i found a really elegant solution in David Morin : View attachment 342284
The idea is that you do these problems yourself. Not just to find a solution online.
 
  • #25
PeroK said:
The idea is that you do these problems yourself. Not just to find a solution online.
This was a new learning experience for me , so i don't really mind . But i do agree that one should try to solve problems on their own . At the end , it just matters that i learnt to solve it even if i had to look at the solution.
I just learnt a new technique and i am happy!

Also , i would love to thank you for patiently helping me throughout the problem 😇
 
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