Pendulums | Oscillations and mechanical waves | Physics | Khan Academy

A pendulum is a mass (m) on a string of length (L) that swings back and forth. It oscillates like a simple harmonic oscillator, making it a key example in studying this phenomenon. The video focuses on the 'simple pendulum' due to its straightforward mathematical description, unlike more complex systems like a double pendulum that exhibit chaotic behavior.

Similar to how an 'X' variable describes linear simple harmonic motion, an angle (theta) can describe a pendulum's angular displacement. The equation for a pendulum's angular position over time (theta(t)) is theta_max * cos(2π/T * t), where theta_max is the maximum angular displacement and T is the period. It's crucial to note that a simple pendulum is only an ideal simple harmonic oscillator for small angles, typically less than 20 degrees, where deviations are minimal.

One might incorrectly assume that increasing the mass of the pendulum would change its period. While a larger mass means more inertia (making it harder to move), it also means a greater gravitational force acting on it (making it accelerate faster). These two effects precisely cancel each other out, meaning the mass of the pendulum does not affect its period. Therefore, two pendulums with different masses but the same length will have the same period.

The period (T) of a simple pendulum is given by the formula T = 2π * sqrt(L/g), where L is the length of the string and g is the acceleration due to gravity. Increasing the length (L) increases the period because it increases the moment of inertia (rotational inertia), making it more difficult to angularly accelerate. Conversely, increasing 'g' (e.g., on a more massive planet) decreases the period because it leads to a stronger restoring force, resulting in faster movement and a shorter cycle time.

The pendulum's period formula shares similarities with a mass-on-a-spring system (T = 2π * sqrt(m/k)). Both equations feature an inertia term on top (L for pendulum, m for spring) and a force-related term on the bottom (g for pendulum, k for spring). In both cases, amplitude does not affect the period, provided the amplitude is small. For pendulums, this 'small angle approximation' generally holds true for angles less than 20 degrees (less than 1% error), and even up to 70 degrees with less than 10% error.