Oscillations: Two General Approaches Someone hands you a system that oscillates, and asks you to ﬁnd the equation of motion. What are you going to do? Well, here are two methods that I ﬁnd useful for solving these things: Forces • Draw a picture of the system • Identify the oscillating object and draw a Free Body Diagram (FBD) for it • Write out Σ F = m a for the object • Rearrange this equation to obtain a diﬀerential equation to your liking • Propose a trial solution ( x ( t )) to the equation, something that oscillates -you expect it to have a cos ( ωt + δ ) piece • Run your solution through the diﬀerential equation - this will usually give you values for ω , and sometimes other constants • Evaluate your solution at some time (usually t = 0) where you have some information about displacement, speed, etc. in order to ﬁnd solutions for the other constants (typically the amplitude and phase shift) Energy (Most useful if W NC = 0 ) • Draw a picture of the system
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