I Effect of Nearby Mountain on an Ideal Pendulum

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The discussion focuses on the effect of a nearby mountain on an ideal pendulum due to a horizontal gravitational force component of 10^-5g. To analyze this, one approach is to determine an angle that modifies the gravitational acceleration to g(-1z + 10^-5x), centering around a symmetric axis. For a first-order approximation, the pendulum can be treated as a standard 1g pendulum with its center of swing displaced by 10^-5 times the length of the pendulum arm (A). This method allows for the incorporation of the perturbation while maintaining the integrity of the ideal pendulum model. The conversation highlights the mathematical transformations necessary to account for the mountain's influence on pendulum dynamics.
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Suppose there is a very large mountain adjacent to a pendulum such that there is a horizontal component gravitational force of ##10^{-5}g## acting on the otherwise ideal pendulum. How would one use a perturbation to add that effect to first order?

My initial thought would be to figure an angle that made the total gravitational acceleration ##g(-1z+10^{−5}x)## operating around a symmetric axis and which gives an ideal pendulum and then transform the coordinates?
 
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If you are only looking for a 1st order approximation, treat it as a 1g pendulum with the center of the swing displaced 10^-5 A were A is the length of the pendulum arm.
 
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