The periodic nature of SHM and simple pendulum  Simple Harmonic Motion (SHM)
The periodic nature of SHM and simple pendulum
Lessons
Notes:
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 toandfro 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 $f =$ $\large \frac{1}{2 \pi}$ $\sqrt{\frac{g}{L}} \enspace K = \frac{mg}{L}$

Intro Lesson

2.
A 6 kg block is attached to a spring wit a spring constant of 216 N/m. The spring is stretched to a length of 12cm and then released.

3.
A mass of 2.40 kg is attached to a horizontal spring with a spring constant of 121N/m. It is stretched to a length of 10.0cm and released from test.

4.
A mass is attached to a horizontal spring, and oscillates with a period of 1.4s and with an amplitude of 12cm. At $t=0$s, the mass is 12cm to the right of the equilibrium positon.

5.
A simple Pendulum has a length of 42.0cm and makes 62.0 complete oscillation in 3.0 min.

6.
The length of a simple pendulum in 0.86m, the pendulum bob has a mass of 265 g and it is released to an angle of 11.0° to the vertical.