Question

Why is the path length twice the amplitude in a simple pendulum?

Answer 1

In a simple pendulum, the path length is twice the amplitude due to the nature of its oscillatory motion. The key factor here is the geometry of the pendulum's motion.

The amplitude of a pendulum refers to the maximum angle it swings away from its equilibrium position (the vertical position when it's at rest). Let's denote the amplitude as "A."

Now, consider the motion of the pendulum as it swings back and forth. When the pendulum reaches its maximum displacement on one side, it has traveled a distance equal to the length of the arc it sweeps through. Let's call this distance "d."

The key insight here is that the pendulum's motion follows an arc of a circle, and the length of this arc is directly related to the amplitude. For small angles (typically less than 15 degrees), the motion of a simple pendulum closely approximates that of a simple harmonic oscillator, and the relationship between the arc length (d) and amplitude (A) can be approximated using trigonometry:

$d\approx A$

Now, let's consider the full motion of the pendulum. It swings from one extreme point (amplitude A) to the other extreme point (amplitude -A) and back. The total distance covered during one complete back-and-forth oscillation is twice the amplitude.

So, the path length (L) covered by the pendulum during one complete oscillation is given by:

$L=2\times d=2\times A$

Therefore, the path length is twice the amplitude in a simple pendulum.