Solve problems involving the simple pendulum

Understanding Simple Pendulum

A simple pendulum is an idealised model consisting of a point mass (the bob) suspended from a fixed point with a weightless string without friction.
The bob of the pendulum executes a type of motion called simple harmonic motion (SHM) when displaced from its equilibrium position.
The equilibrium position is the vertical position the pendulum attains when it’s at rest.
The displacement of the pendulum is the arc length from the equilibrium position to its current position.
The pendulum experiences a restoring force proportional to its displacement and directed towards the equilibrium position.

The length of the pendulum (distance between the fixed point and the centre of the bob) can heavily influence its motion.
The period of a simple pendulum, representing the time for one complete swing (back and forth), is independent of the amplitude (provided the angle of swing is small) and depends only on the length of the pendulum and acceleration due to gravity.
The formula for the period T of a simple pendulum is given by
```
  T = 2π sqrt(l/g)
```
where l is the length of the pendulum and g is the acceleration due to gravity.

Start by sketching out a diagram of the pendulum for better clarity of the forces involved and the direction of displacement.
Identify the key parameters in the problem: the length of the pendulum, the angle of swing, and the time taken for a complete swing or part thereof.
Use the formula for the period of a simple pendulum to solve for unknowns when two quantities are given.
Remember that the period does not change with amplitude or mass of the bob (for small angles of swing).
Be mindful of units while substituting values into formulae and performing calculations.
When dealing with angles, it might be useful to apply small angle approximations where it is presumed that sin θ ≈ θ and cos θ ≈ 1 when θ is small and measured in radians.
Always offer a physical or intuitive explanation, where possible, alongside the mathematical solution.

When an external force or mechanism dissipates the energy of the oscillator, the pendulum experiences damping.
Damping causes the amplitude of the pendulum to decrease gradually, slowing down the motion.
While the specific effects vary with the nature of the damping, common forms of damping in pendulum include air resistance and friction at the pivot point.
Understanding the concept of damping is important when analysing the real-world behaviour of pendulums.