1. Start with rotational Newton's Second Law. 2. Define the torque equation: $\tau = -mgd \sin\theta$. 3. Apply small angle approximation: $\sin \theta \approx \theta$. 4. Relate torque and angular acceleration: $\tau = I\alpha$. 5. Calculate angular acceleration. 6. Use $T = \frac{2\pi}{\omega}$ to find the period: $T_{\text{phys}} = 2\pi \sqrt{\frac{I}{mgd}}$

1. Divide the object into simpler shapes. 2. Determine the moment of inertia of each shape. 3. Use the parallel axis theorem if necessary to shift the axis of rotation. 4. Sum the moments of inertia of all the shapes to find the total moment of inertia.

1. Identify the initial and final states of the pendulum. 2. Determine the potential energy (gravitational) and kinetic energy (rotational) at each state. 3. Apply the conservation of energy principle: $U_i + K_i = U_f + K_f$. 4. Solve for the unknown variable (e.g., angular velocity or maximum angle).

1. Recognize that the angle of displacement is small. 2. Replace $\sin \theta$ with $\theta$ (in radians). 3. Use the simplified equations for torque and angular acceleration. 4. Solve for the period or angular frequency using the simplified expressions.

1. Apply the small-angle approximation to linearize the motion. 2. Show that the angular acceleration is proportional to the angular displacement: $\alpha = -\omega^2 \theta$. 3. Recognize that this is the condition for SHM. 4. Use the SHM equations to analyze the motion of the pendulum.

Simple Pendulum: Point mass on a massless string, period depends only on length. | Physical Pendulum: Rigid object with distributed mass, period depends on moment of inertia and mass distribution.

Torque: Rotational force, causes angular acceleration, depends on the distance from the pivot point. | Force: Linear force, causes linear acceleration, independent of pivot point.

Angular Velocity: Rate of change of angular displacement, measured in radians per second, describes rotational motion. | Linear Velocity: Rate of change of linear displacement, measured in meters per second, describes translational motion.

Moment of Inertia: Resistance to rotational motion, depends on mass distribution, $I = \sum mr^2$. | Mass: Resistance to linear motion, scalar quantity, intrinsic property of an object.

Potential Energy: Energy due to position, $U = mgh$, maximum at maximum displacement. | Kinetic Energy: Energy due to motion, $K = \frac{1}{2}I\omega^2$, maximum at equilibrium position.

A rigid object that swings back and forth around a fixed pivot point, with a complex shape and mass distribution.

A measure of an object's resistance to changes in its rotational motion, dependent on mass distribution.

A rotational force that causes an object to rotate around an axis.

The rate of change of an angle, measured in radians per second, representing how quickly an object oscillates.

A type of periodic motion where the restoring force is directly proportional to the displacement, resulting in sinusoidal oscillations.