A pendulum is a ball m hanging from a massless string of length L and fixed with a pivot point.
If the ball is moved to angle µ and released, it will be moving back and forth in a circular motion.
Applying Newton's Second Law of Motion, the motion equation for the pendulum can be found.
Newton's 2nd Law of Motion (translational): F = m . a
Newton's 2nd Law of Motion (rotational): 𝜏 = I . 𝛼
In this case, we are interested in the rotational movement.
Newton's 2nd Law of Motion for Rotation: 𝜏 = I . 𝛼
=> - m . g . sin(µ) . L = m . L2 . d2(µ) / d(t2)
=> g * sin(µ) / L + d2(µ) / d(t2) = 0
This equation indicates that the simple pendulum would be moving in simple harmonic motion.
Where θ0=µ and w = √(g/L), the simple harmonic equation is θ(t) = θ0 . cos(w . t)
Based on the fact that the simple pendulum moves in a simple harmonic motion, the period of the motion can be calculated.
As previously mentioned, the simple harmonic equation is θ(t) = θ0 . cos(w . t)
Since w = 2 . π / w,
Period of the motion: T = 2 . π . √(L/g)
When the pendulum is released, it will swing back to its initial position.
But the motion will not end here.
If a simple pendulum is released from an initial angle, it will keep moving back and forth in this simple harmonic motion with period T.
Kinetic energy of the pendulum can be calculated as follows: KE = 1/s . m . v2
Gravitational potential energy of the pendulum can be calculated as follows: PE = m . g . h
As the pendulum swings from center (equilibrium) to the left and right, the KE and PE changes.