With good sketch, derive the equation of motion of a compound pendulum. What is the difference between a simple pendulum and a compound pendulum? State the equation(s) for calculating natural frequency in both pendulums.

The only force creating a torque about the pivot O is gravity, $mg$, acting at the center of mass G.
The torque $\tau$ due to gravity about the pivot O is given by: $\tau = -(mg)(L \sin\theta)$ The negative sign indicates that the torque acts to restore the pendulum to its equilibrium position (opposite to the direction of increasing $\theta$).

    Substituting this approximation into the equation:
    $$
    I\ddot{\theta} + mgL\theta = 0
    $$
    This is the equation of motion for a compound pendulum under small oscillations.
    $$
    \boxed{I\ddot{\theta} + mgL\theta = 0}
    $$

2. Difference between a Simple Pendulum and a Compound Pendulum * A simple pendulum consists of a point mass (bob) suspended by a massless, inextensible string from a fixed pivot. All its mass is concentrated at a single point. * A compound pendulum (or physical pendulum) is any rigid body that oscillates about a fixed pivot point under the influence of gravity. Its mass is distributed throughout its volume, and its moment of inertia about the pivot is a significant factor in its motion.

3. Equations for Calculating Natural Frequency

   • Simple Pendulum: The natural angular frequency $\omega_n$ for a simple pendulum of length $L$ is:

     $$\boxed{\omega_n = \sqrt{\frac{g}{L}}}$$

     where $g$ is the acceleration due to gravity.

   • Compound Pendulum: The natural angular frequency $\omega_n$ for a compound pendulum is:

     $$\boxed{\omega_n = \sqrt{\frac{mgL}{I}}}$$

     where $m$ is the total mass of the pendulum, $g$ is the acceleration due to gravity, $L$ is the distance from the pivot to the center of mass, and $I$ is the moment of inertia of the pendulum about the pivot point.