In this lesson, we will learn:

- The nature of the periodic motion
- The graph analysis of the periodic motion
- Simple Pendulum

__Notes:__- If each vibration (the back and forth motion) takes the same amount of time, then the motion is Periodic.
- To discuss the period motion we need to define the following terms;
**Cycle;**a complete to-and-fro movement.**Amplitude;**maximum displacement, the greatest distance from the equilibrium position.**Period;**time taken for one complete oscillation.- Frequency; number of oscillations in one second.

- The following equations represent the mathematical relationship between frequency and period of motion;

**Position as a Function of Time**

Since the motion is considered as a periodic motion, we would be able to plot position Vs. time graph.

Looking at the graph we can refer to it as the cosine function, since at t = 0 the position is maximum;

$w =$ angular velocity

$w = 2 \pi f = 2 \pi / T$

- For one complete cycle,

**Velocity as a Function of Time**

- Velocity is defined as the derivative of position with respect to time,

Since the Velocity is the sine function of time, at $t = 0, V = 0$, therefore; the graph starts at zero.

**Acceleration as a Function of Time**

- Acceleration is defined as the derivative of velocity with respect to time,

**Simple Pendulum**

- A simple pendulum is a small mass attached to the end of a string.
- The pendulum swings back and forth, ignoring the air resistance, it resembles simple harmonic motion.

Let’s apply the simple harmonic oscillator to the case of the simple pendulum;

In the case of simple pendulum;

$A$; is the maximum angular displacement, $\theta _{max}$

$x$; is the angle the pendulum is at; the angle is measured from the equilibrium positon (the vertical position),$\theta$.

- The restoring force is opposite to the displacement and is equal to the component of the weight;
- In this case, the motion is considered to be simple harmonic motion if the angle is less than 15°, for small angles, $\sin \theta \approx \theta$;

Form Hooke’s law;

We know from spring- mass system;

$T = 2 \pi \, \sqrt{\frac{m}{k}} \, \Rightarrow \, T = 2 \pi \, \sqrt{\frac{m}{mg\, / \, L} } \, \Rightarrow \, T = 2 \pi \, \sqrt{\frac{L}{g}}$

and $f =$ $\large \frac{1}{2 \pi}$ $\sqrt{\frac{g}{L}} \enspace K = \frac{mg}{L}$