Pendulum Using Lagrange And Hamilton

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To determine the period of a pendulum with an upward accelerating support, one must first express the coordinates in polar form and visualize the problem. The kinetic and potential energies need to account for both gravitational force and the upward acceleration. The Lagrangian or Hamiltonian should then be derived to establish the equations of motion. The formula for the period is not simply 2π(L/g-a)^(1/2) but requires a more complex approach, especially when considering extreme cases like upward acceleration equaling g. Understanding these dynamics is crucial for accurately solving the problem.
skrao
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i have been given a problem involving a pendulum, where its support point is accelerating vertically upward. The period of the pendulum is required. Anybody have any idea how to start this one? is it not just 2pi(L/g-a)^1/2?
 
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First thing to do is write x, y and x', y' in terms of r and theta (polar coords). Draw a picture to help you visualize, then find the kinetic and potential energies and be sure to take into account the upwards acceleration, so you have both gravity and this upwards accelration acting on the mass. Then find the Lagrangian and/or Hamiltonian and use one of them to find the equations of motion.
 
The period of a pendulum is not \nu = 2\pi \sqrt{\frac{L}{g}} that is only an approximation to the right expression, which we can't solve exactly
 
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skrao -- Ask yourself what the period would be if the upward acceleration was equal to g?

Regards,
reilly Atkinson
 

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