Stable and Unstable Equilibriums in a Spring-Mass System on a Loop

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In a spring-mass system on a loop, two stable equilibria occur at positions where the mass is at the lowest points of the loop, while two unstable equilibria are found at the top of the loop. The spring's attachment point, fixed at a distance d from the center, influences the stability based on gravitational forces and the spring's tension. The mass can slide along the diameter, but the hoop rolls without slipping, maintaining the system's dynamics. The fixed pivot point of the spring ensures that the distance d remains constant throughout the motion. Understanding these equilibria is crucial for analyzing the system's behavior under various conditions.
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Imagine drawing the following figure: draw a circle and a diameter. They are made of massless wire. There's a spring attached a distance d from the center of the hoop on the diameter, and a mass on the other end of the spring.

Using physical intuition, where would the two stable equibriums and two unstable equilibriums be?



Note:
natural length of spring is 0
the hoops rolls without slipping
straight wire is massless
yes, the mass can slide both ways
each end of the wire is connected to the loop
 
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is the pivot point moveable? (i.e. does d=d(t) or does d=const)
 
No, the pivot point of the spring is fixed.
 

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