Let's work through this problem step by step.

Part 1: Angular Frequency ($\omega$) of the Ideal Pendulum

We will derive the angular frequency using Newton's Second Law and the small-angle approximation.

Step 1: Forces on the pendulum

• The restoring force due to gravity is $F = -m g \sin\theta$, where $m$ is the mass, $g$ is acceleration due to gravity, and $\theta$ is the angular displacement.
• For small angles ($\sin\theta \approx \theta$ in radians), the force becomes: $F \approx -m g \theta$

Step 2: Relating force to torque

The torque about the pivot is: $\tau = -m g l \theta$ where $l$ is the length of the pendulum.

Step 3: Rotational dynamics

The torque is also related to the moment of inertia ($I = m l^2$) and angular acceleration ($\alpha = \ddot{\theta}$): $\tau = I \alpha = m l^2 \ddot{\theta}$ Equating the two expressions for torque: $m l^2 \ddot{\theta} = -m g l \theta$

Step 4: Simplifying the equation

Dividing through by $m l$: $l \ddot{\theta} + g \theta = 0$

Step 5: Angular frequency

This is the equation of simple harmonic motion, where the angular frequency ($\omega$) is: $\omega = \sqrt{\frac{g}{l}}$

Part 2: Period ($T$) of the Pendulum

The period of the pendulum is related to the angular frequency: $T = \frac{2\pi}{\omega}$ Substituting $\omega = \sqrt{\frac{g}{l}}$: $T = 2\pi \sqrt{\frac{l}{g}}$

Final Expressions:

1. Angular frequency: $\omega = \sqrt{\frac{g}{l}}$
2. Period: $T = 2\pi \sqrt{\frac{l}{g}}$

Let me know if you need further clarifications! Here are some follow-up questions to explore:

1. How does the length of the pendulum ($l$) affect the period and angular frequency?
2. What assumptions are made when deriving these equations, and how might they break down?
3. How would the period change if the pendulum were on a planet with different gravitational acceleration?
4. Can you derive the same expressions using energy methods?
5. What happens to the motion if the small-angle approximation ($\sin\theta \approx \theta$) is not valid?

Tip: For accurate small-angle approximation, keep $\theta_{\text{max}}$ below 15° for better precision.