L02_a - Simple and Physical Pendula Simple Pendulum"point...

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Simple and Physical Pendula Simple Pendulum: “point” mass m on massless string L How do we find natural frequency ω ? o Use Newton’s 2 nd law to get differential equation for SHM o Identify ω from differential equation Forces on hanging mass: o For string at θ from vertical, θ sin tangential mg F =
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Tangential acceleration: o Reminder: for point at distance R from centre of body with angular acceleration 2 2 dt d θ α = , tangential acceleration is α R a = tangential o So for mass on string of length L , tangential acceleration is 2 2 tangential dt d L a θ = (Remember: θ must be in radians here) Newton’s 2 nd Law: tangential tangential ma F = gives 2 2 sin dt d mL mg θ θ = o Result: θ θ sin 2 2 L g dt d = o Not quite SHM BUT for small angle, θ θ sin (for θ in radians)
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Simple Pendulum result: for small θ , have θ θ L g dt d = 2 2 o Looks like SHM with L g = 2 ω Note, does not depend on m ! Strictly SHM for small angle only. Slightly different from sinusoidal for larger amplitude.
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