Rotational Force §

Torque §

Torque (or 𝜏) is a measure of the rotational force acting on an object. In the case of a pendulum, torque is proportional to both the mass of the bob and the length of the arm (𝑀 × 𝑟).

𝜏 = 𝑀 × 𝑟

Moment of inertia §

The moment of inertia (or 𝐼) of a pendulum is a measure of the amount of difficulty in rotating the pendulum around the pivot point. It’s proportional to the mass of the bob and the square of the length of the arm (𝑀 x 𝑟2).

𝐼 = 𝑀 x 𝑟2

Rotational counterpart of Newton’s second law §

# Newton’s second law
𝐹 = 𝑀 × 𝐴
 
# Rotational counterpart
𝜏 = 𝐼 × 𝛼

By rearranging the equation to solve for the angular acceleration 𝛼, I get 𝛼 = 𝜏 / 𝐼. Simplifying further, this becomes 𝑀𝑟 / 𝑀𝑟2 or 1 / 𝑟. The angular acceleration doesn’t depend on the pendulum’s mass!

This is just like Galileo’s Leaning Tower of Pisa experiment demonstrating linear acceleration, where different objects fell at the same rate, regardless of their mass. Here, once again, the mass of a bob doesn’t influence its angular acceleration—only the length of its arm does.

Reference §

3. Oscillation / Nature of Code