How Do You Calculate the Spring Constant in a Coupled Pendulum System?

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To calculate the spring constant in a coupled pendulum system, the period of the pendulum when one bob is fixed provides crucial information. The period of 1.25 seconds indicates the system's dynamics, but the spring constant k cannot be directly derived from this period alone. Analyzing the forces acting on the movable bob, both gravitational and spring forces must be considered to determine the total restoring force. By applying principles of harmonic motion and the relationship between frequency, mass, and spring constant, k can be calculated. Understanding these dynamics is essential for finding the periods of the normal modes when both bobs are free.
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I am given a set up with two pendulums of unknown mass m, of length =.4 meters. They are connected together with a spring of unknown spring constant k. It says when one of the bobs if fixed in place the other has a period of 1.25 seconds. I am then asked to find the period of each normal mode when both bobs are free. I know I need to find k but don't understand how using the information given about the pendulum with a spring attached. I know the frequency of the pinned system but am not sure how to get k from that, as it isn't just equal to k/m.
 
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Take the movable bob and displace it by some small distance dx from its equilibrium position. Now what is the total restoring force on it (from gravity + spring) ?
 
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