Pendulum in accelerating traincar

AI Thread Summary
The discussion revolves around determining the oscillatory period of a pendulum in a constantly accelerating traincar using the small angle approximation. The equations of motion are derived, incorporating gravitational and acceleration forces acting on the pendulum bob. A differential equation is formed, combining the effects of gravity and acceleration. The original poster seeks the most efficient method to solve this differential equation, contemplating the use of Fourier transforms. The conversation emphasizes finding a straightforward solution to the problem.
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Homework Statement


A pendulum bob of mass m is suspended by massless thread of length l from the ceiling of a boxcar. The car undergoes constant acceleration a along a straight, level track. What, in the small angle approximation, is the oscillatory period tau of the pendulum?

Homework Equations


F_g=mg\sin\theta=mg\theta
F_a=ma\cos\theta=ma

The Attempt at a Solution


mg\theta+ma=ml\ddot{\theta}
\frac{g}{l} \theta+\frac{a}{l}=\ddot{\theta}

What's the quickest way to solve this differential equation?
 
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My first idea was to Fourier transform it, but I'm not sure if that would be the best approach. Any suggestions?
 

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