Coupled pendulum-spring system

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The discussion centers on the separation vector in a coupled pendulum-spring system, specifically how to express it algebraically. The user seeks clarification on the notation and the relationship between the angles and displacements of the masses. Participants suggest that for small perturbations, the unit vectors can be simplified to a single direction, ##\hat{k}##. They explain that the displacement can be treated as equivalent regardless of the order of subtraction due to the nature of small angles. The conversation emphasizes understanding the geometric relationships in the system for accurate modeling.
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Homework Statement
Please see below. I have a doubt about the solution of the problem.
Relevant Equations
Arc length = ##R\theta##
The problem and solution are,
1712980472532.png

1712980490956.png

1712980508579.png

However, I am confused how the separation vector between the two masses is

##\vec x = x \hat{k} = x_2 \hat{x_2} - x_1 \hat{x_1}= l\theta_2 \hat{x_2} - l\theta_1 \hat{x_1 } = l(\theta_2 - \theta_1) \hat{k}##. where I define the unit vector from mass 2 to mass 1 as ##\hat{k}## along the spring. Does someone please know of a algebraic or geometric proof for this?

Thanks alot!
 
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ChiralSuperfields said:
Homework Statement: Please see below. I have a doubt about the solution of the problem.
Relevant Equations: Arc length = ##R\theta##

The problem and solution are,
View attachment 343276
View attachment 343277
View attachment 343278
However, I am confused how the separation vector between the two masses is

##\vec x = x \hat{k} = x_2 \hat{x_2} - x_1 \hat{x_1}= l\theta_2 \hat{x_2} - l\theta_1 \hat{x_1 } = l(\theta_2 - \theta_1) \hat{k}##. where I define the unit vector from mass 2 to mass 1 as ##\hat{k}## along the spring. Does someone please know of a algebraic or geometric proof for this?

Thanks alot!
Which particular step worries you?
It seems a bit verbose to me. Why introduce ##\hat{x_2} ## and ## \hat{x_1}##? Aren’t they obviously ##\hat k## for small perturbations (compared to the length of the spring)?
 
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haruspex said:
Which particular step worries you?
It seems a bit verbose to me. Why introduce ##\hat{x_2} ## and ## \hat{x_1}##? Aren’t they obviously ##\hat k## for small perturbations (compared to the length of the spring)?
Thank you for your reply @haruspex!

Sorry are you saying that ##\hat{x_2} = \hat{x_1} = \hat{k}## for small ##\theta##? My confusion is how the spring displacement from equilibrium position is ##l\theta_2 - l\theta_1 = l\theta_1 - l\theta_2## since the displacement is squared. I am confused when I try to imagine subtracting two arc lengths from each other.

Thanks!
 
ChiralSuperfields said:
My confusion is how the spring displacement from equilibrium position is ##l\theta_2 - l\theta_1 = l\theta_1 - l\theta_2## since the displacement is squared. I am confused when I try to imagine subtracting two arc lengths from each other.

Thanks!
For small perturbations, the arc length is near enough the same as the displacement in the initial tangential direction, i.e. the ##\hat k## direction.
Since it is squared, it does not matter which way around you do the subtraction.
 
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