Coupled pendulum-spring system

[ChiralSuperfields]

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,

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!

 

[haruspex]

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)?

 

[ChiralSuperfields]

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!

 

[haruspex]

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.