The small-angle approximation for the frequency $\omega$ of a simple pendulum can be derived using the following steps:

1. Equation of Motion: The equation of motion for a simple pendulum is given by: $\frac{d^2\theta}{dt^2} + \frac{g}{L} \sin(\theta) = 0$ where:

   • $\theta$ is the angular displacement,
   • $g$ is the acceleration due to gravity,
   • $L$ is the length of the pendulum.

2. Small-Angle Approximation: For small angles (where $\theta$ is in radians), $\sin(\theta) \approx \theta$. Thus, the equation simplifies to: $\frac{d^2\theta}{dt^2} + \frac{g}{L} \theta = 0$ This is a simple harmonic oscillator equation of the form: $\frac{d^2\theta}{dt^2} + \omega^2 \theta = 0$ where $\omega$ is the angular frequency.

3. Angular Frequency: Comparing the simplified equation with the standard form, we find: $\omega^2 = \frac{g}{L}$ Thus, the angular frequency $\omega$ is: $\omega = \sqrt{\frac{g}{L}}$

So, the small-angle approximation for the angular frequency of a simple pendulum is $\omega = \sqrt{\frac{g}{L}}$.

Would you like more details on any of these steps or have any other questions?

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Here are 8 related questions you might find interesting:

1. How does the length of the pendulum affect the frequency?
2. What is the physical interpretation of angular frequency $\omega$?
3. How does the small-angle approximation break down at larger angles?
4. What are the differences between angular frequency $\omega$ and linear frequency $f$?
5. How would the frequency change if the pendulum were on the Moon?
6. Can we use the small-angle approximation for a conical pendulum?
7. What role does damping play in the frequency of a pendulum?
8. How can the period of a pendulum be derived from the angular frequency?

Tip: When working with pendulum problems, always check if the small-angle approximation is valid before applying it.