Find the natural frequency of a system of two pendulums

[joker_900]

Homework Statement

OK I haven't been given a definite question with values, but how would you find the natural frequency of a system of two pendulums coupled by a spring?

Homework Equations

a= -gx/l
T=2pi sqrt(l/g)

The Attempt at a Solution

Well the thing is, I know I need to consider each pendulum/spring separately and get two equations of motion, but I'm not sure how. All i know about pendulums are those two equations above. When I've had similar problems with springs I've been able to find the resultant force as a function of x and go from there. But in this case I just came up with the resultant -mgx/l - lamda(x-0.5l) = ma

Help!

 

[physics girl phd]

I know I need to consider each pendulum/spring separately and get two equations of motion, but I'm not sure how.

Some thoughts:
Have you drawn out a picture of the system? Are the pedulums fixed at the same point at the top or are their top points separated by the natural length of the spring?

Then: Can you use free body diagrams to find out the forces on the bobs of the pendulums? What if you displace one bob in the system by a small angle?

 

[m2003]

To find the natural frequency of a system of two pendulums coupled by a spring, you can use the concept of normal modes. This involves finding the normal modes of the system, which are the modes of oscillation that occur when the system is in equilibrium. These modes are characterized by a specific frequency, known as the natural frequency, which is determined by the properties of the system.

To find the normal modes of the system, you can use the equations of motion for each pendulum and the spring. These equations can be written in the form of a matrix equation, where the variables represent the displacement of each pendulum and the spring from their equilibrium positions. By solving this matrix equation, you can determine the normal modes and their corresponding frequencies.

Alternatively, you can use the concept of energy to find the natural frequency. The total energy of the system is equal to the sum of the kinetic and potential energies of each pendulum and the spring. By setting the derivative of the total energy with respect to time equal to zero, you can find the frequencies at which the system will oscillate.

In summary, to find the natural frequency of a system of two pendulums coupled by a spring, you can use the concept of normal modes or the concept of energy. Both approaches will lead to the same result, which is the natural frequency of the system.