C.1.1 Defining and analyzing simple harmonic motion Notes

Conditions for Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is a special type of oscillatory motion where an object moves back and forth around an equilibrium position.

Restoring Force Proportional to Displacement

1. For SHM to occur, the system must have a restoring force that:
   1. Acts in the opposite direction to the displacement.
   2. Is directly proportional to the displacement from the equilibrium position.
2. This relationship can be expressed mathematically as: $$
   F = -kx
   $$ where:
   1. $F$ is the restoring force.
   2. $k$ is a constant (often called the spring constant in a spring-mass system).
   3. $x$is the displacement from the equilibrium position.

Acceleration in SHM

1. The restoring force causes the object to accelerate back towards the equilibrium position.
2. Using Newton’s second law, $F = ma$, we can express the acceleration as: $$
   a = -\frac{k}{m}x
   $$
3. This shows that the acceleration is also proportional to the displacement and acts in the opposite direction.

Defining Equation for SHM: $a = -\omega^2 x$

The defining equation for SHM is:

$$
a = -\omega^2 x
$$

where:

1. $a$ is the acceleration.
2. $x$ is the displacement.
3. $\omega$ (omega) is the angular frequency, a constant that characterizes the system.

Why $a = -\omega^2 x$?

1. The term $\omega^2$ is derived from the system’s properties.
2. In a mass-spring system, for example, $\omega^2 = \frac{k}{m}$, where $k$ is the spring constant and $m$ is the mass.

This relationship highlights how the system’s physical characteristics determine its oscillatory behavior.

Key Parameters of SHM

To fully describe SHM, we need to understand several key parameters:

Amplitude, Equilibrium Position, and Displacement

1. Amplitude ($A$):
   1. The maximum displacement from the equilibrium position.
   2. It represents the furthest point the object reaches during its oscillation.
2. Equilibrium Position:
   1. The point where the net force on the object is zero.
   2. In SHM, the object oscillates symmetrically around this position.
3. Displacement ($x$):
   1. The distance of the object from the equilibrium position at any given time.
   2. Displacement can be positive or negative, depending on the direction.

Time Period ($T$), Frequency ($f$), and Angular Frequency ($\omega$)

These parameters describe the timing of the oscillation:

1. Time Period ($T$): The time taken to complete one full oscillation. It is measured in seconds (s).
2. Frequency ($f$): The number of oscillations per second. It is measured in hertz (Hz) and is the inverse of the period: $$
   f = \frac{1}{T}
   $$
3. Angular Frequency ($\omega$): A measure of how quickly the object oscillates, expressed in radians per second (rad/s). It is related to the period and frequency by: $$
   \omega = 2\pi f = \frac{2\pi}{T}
   $$