What is the Angular Frequency of Two Masses on a Spring?

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To calculate the angular frequency of two masses attached to a spring, the equations of motion for each mass must be considered, specifically focusing on the relationship between their positions. The center of mass (CM) plays a crucial role in simplifying the equations, allowing for a single equation that relates the motion of both masses. The suggestion to combine the equations into one that depends on the second derivative of the distance between the two masses is a valid approach. Understanding how each mass moves relative to the center of mass is essential for deriving the angular frequency. The discussion emphasizes the importance of correctly formulating the equations to find the solution.
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Homework Statement


We attach two blocks of masses m1 = 10 kg and m2 = 4 kg to either end of a spring of spring constant k = 4 N/m and set them into oscillation. Calculate the angular frequency ω of the oscillation

Homework Equations


M_{1}{x}^{..}=k(w-x)
M_{2}{w}^{..}=-k(w-x)

where x and w are the distance from equilibrium position of m1 and m2 respectively.

Attempt

Honestly, I have absolutely no idea. How does one start this?
 
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Notice that the right-hand sides depend only on w-x. Wouldn't it be nice if you had an equation where the left-hand side depended only on (d/dt)2(w-x)? How could you get such an equation?
 
What can one say about the center of mass? How does each move with respect to the CM?

Also use \ddot{x} for \ddot{x}
 
Are you suggesting I make one equation for both masses instead of two separate equations?

x_{cm}=\frac{{m_{1}x_{1}+m_{2}x_{2}}}{{\sum{m}}}

x_{1}=w, x_{2}=x
 
I know this thread is old, but I was wondering if someone could confirm the attempt suggested in the above post.
 

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