Harmonic oscillator and simple pendulum time period

In summary, a simple pendulum will take the same time to complete one period, no matter which length it has, if stiffness and mass are the same. But, for a simple pendulum, with small angle approximation the period changes with its length.
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The force applied to the mass is the same no matter how long it is (at least for small angles), so the tangential velocity is the same too. The angular velocity is not the same because of:

w = v/L

The angular velocity for a longer L is less, and for a shorter L more. You need more time to pass the same angle with longer L because arc length is longer.
 

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