Are the Force Equations for Rotational Motion Accurate?

  • #1
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 : and for block : 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 where
 
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  • #2
Have you done any Lagrangian mechanics?
 
  • #3
PeroK said:
Have you done any Lagrangian mechanics?
No
 
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  • #4
Bling Fizikst said:
No
It should come out from conservation of AM and energy. You're aiming for an equation for . Which you should be able to get from:

Bling Fizikst said:
where
 
  • #5
... differentiating the energy equation should work.
 
  • #6
PeroK said:
... differentiating the energy equation should work.
I did that but how do i deal with the potential energy of the system :
 
  • #7
Bling Fizikst said:
I did that but how do i deal with the potential energy of the system :
You should be able to express that in terms of .
 
  • #8
PeroK said:
You should be able to express that in terms of .
Need help on this , really deadstuck . Writing forces and using doesn't seem to help?
 
  • #9
Bling Fizikst said:
I did that but how do i deal with the potential energy of the system :
What precisely is stopping you from writing down an expression for ?
 
  • #10
Bling Fizikst said:
Need help on this , really deadstuck . Writing forces and using doesn't seem to help?
How does the position of the mass M depend on ? Take as the zero point.
 
  • #11
Is it -Mgr:nb)
 
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  • #12
Bling Fizikst said:
Is it -Mgr:nb)
What happens to as 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 is related to by the initial equilibrium.
 
  • #15
 
  • #16
Bling Fizikst said:
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.
implies .
 
  • #18
vela said:
implies .
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 :
I wonder whether you replaced with here? To get from the above equation to this:
Bling Fizikst said:
In any case, the solution only applies to small oscillations. You will need an approximation somewhere.
 
  • #21
PeroK said:
I wonder whether you replaced with here? To get from the above equation to this:
Yes
 
  • #22
Note that is a constant, like and and is the fixed angular velocity at the equilibrium. and are the variables. We have:
Also, with , we have:
And
 
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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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