Solve equations using Simple Harmonic motions

Basic Concepts of Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. This results in an oscillatory motion.
An object in SHM moves back and forth within a stable equilibrium position under the action of a restoring force proportional to its displacement.

The Nature of SHM

The amplitude of the SHM refers to the furthest point from the equilibrium position that the object reaches. It’s the maximum displacement.
Period is the time taken for one complete cycle of motion to repeat itself.
The frequency of the SHM is simply the reciprocal of the period.

Equations of Simple Harmonic Motion

The displacement x as a function of time t for Simple Harmonic Motion can be written as:
```
  x = A cos(wt + ø),
```
where A is the amplitude, w is the angular frequency, and ø is the phase angle.
The velocity and acceleration in SHM can be derived from the displacement equation.
```
  v = - A w sin( w t + ø ),

  a = - A w^2 cos( w t + ø )
```

Solving SHM Problems

To solve most simple harmonic motion problems, begin by identifying the amplitude, period, frequency, and phase shift.
Substituting time t at different points in the period (like when one cycle begins and ends) to the SHM equations can often present useful insights.
Equate expressions obtained to model conditions given in statement of problem, to solve for constants such as amplitude A, angular velocity w, and phase shift ø .
Remember that SHM velocity reaches a maximum at equilibrium and is zero at extremes; acceleration is zero at equilibrium and maximum at extremes.

Energy in SHM

The total energy in SHM is a sum of kinetic and potential energies. It’s constant if no damping acts on it.
Consider the potential and kinetic energy transformations when dealing with problems involving energy. They are maximum and minimum at opposite instances.

Note: While practising, remembering to visualise the harmonic motion can significantly assist in understanding the problem better. It’s essential to keep in mind that the restoring force seeks to bring the object back to equilibrium whenever it strays.