# Discover the Periodic Nature of SHM and Simple Pendulums Dive into the world of periodic motion! Understand simple harmonic motion and simple pendulums through engaging visuals and real-world applications. Master these fundamental concepts in physics and mathematics.

Now Playing:The periodic nature of shm and simple pendulum1 – Example 0a
Intros
1. Periodic Motion
2. Position Vs. Time Graph
Examples
1. The spring of a 1200kg car compresses 2.0mm when its 75kg driver gets into the car. If the car goes over a bump, what will be the frequency of the vibration?
Vibration and energy
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 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;

$\large f = \frac{1}{T} \quad$ and $\quad\large T = \frac{1}{f}$

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;

$x \, (t) = Acoswt$

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

$x \, (t) = Acos(2 \pi f t) = Acos$ $\large (\frac{2 \pi t}{T})$

• For one complete cycle,

Velocity as a Function of Time

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

$v =$ $\large \frac{dx}{dt}$ $=$ $\large \frac{d}{dt}$ $(Acoswt) = - Aw (sinwt)$

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,

$a =$ $\large \frac{dv}{dt}$ $=$ $\large \frac{d}{dt}$ $(-Awsinwt) = - Aw^{2} \, (coswt)$

$a =$ $\large \frac{F}{m}$ $= -$ $\large \frac{kx}{m}$ $=-$ $\large (\frac{kA}{m})$ $coswt = - a_{max} coswt$

$\Rightarrow a_{max} =$ $\large \frac{kA}{m}$

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;

$x(t) = Acos$$\large (\frac{2 \pi}{T} \, t)$

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),$\theta$.

$\theta \, (t) = \theta_{max} cos$ $\large (\frac{2 \pi}{T}\, t)$

• The restoring force is opposite to the displacement and is equal to the component of the weight;

• $F = - mgsin \theta$

• 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;
$F = - Kx$
$\qquad \qquad \qquad \qquad \qquad \qquad \Rightarrow K =$ $\large \frac{mg}{L}$

$F \approx -$ $\large \frac{mgx}{L}$

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

## Introduction to Periodic Motion: SHM and Simple Pendulum

Welcome to our exploration of periodic motion! Today, we'll dive into two fascinating concepts: simple harmonic motion (SHM) and the simple pendulum. These are fundamental to understanding many natural phenomena and mechanical systems. To kick things off, I've prepared an introduction video that will give you a visual grasp of these concepts. This video is crucial as it demonstrates the key principles we'll be discussing. In simple harmonic motion, an object oscillates back and forth around an equilibrium position. The simple pendulum is a perfect example of SHM in action. As we progress, you'll see how these concepts apply to real-world scenarios, from the ticking of a grandfather clock to the vibrations in musical instruments. The beauty of these concepts lies in their predictability and mathematical elegance. So, let's get started on this exciting journey through the world of periodic motion!

FAQs
1. #### What is the difference between periodic motion and simple harmonic motion?

Periodic motion is any motion that repeats itself at regular intervals. Simple harmonic motion (SHM) is a specific type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position. All SHM is periodic, but not all periodic motion is SHM. For example, a pendulum swinging with small amplitudes exhibits SHM, while a planet orbiting the sun is periodic but not SHM.

2. #### How does the period of a simple pendulum change with its length?

The period of a simple pendulum is directly proportional to the square root of its length. This relationship is described by the equation T = 2π(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. If you double the length of a pendulum, its period will increase by a factor of 2, or about 1.41 times.

3. #### Why doesn't the mass of the bob affect the period of a simple pendulum?

The mass of the bob doesn't appear in the equation for the period of a simple pendulum (T = 2π(L/g)) because the gravitational force and the inertia of the bob both increase proportionally with mass. These effects cancel each other out, resulting in the period being independent of mass. This is true as long as the pendulum's motion is not significantly affected by air resistance.

4. #### What are some real-world applications of simple harmonic motion?

Simple harmonic motion has numerous applications in everyday life and technology. Some examples include: - Timekeeping devices like pendulum clocks - Musical instruments (strings and wind instruments) - Seismographs for detecting earthquakes - Shock absorbers in vehicles - Design of earthquake-resistant buildings - Spring-based weighing scales These applications utilize the predictable and repeatable nature of SHM to measure time, create sound, or absorb and dissipate energy.

5. #### How does air resistance affect the motion of a simple pendulum?

In ideal conditions, a simple pendulum would continue to oscillate indefinitely. However, in reality, air resistance gradually reduces the amplitude of the pendulum's swing, causing it to eventually come to rest. This effect is more pronounced for pendulums with larger surface areas or those swinging at higher speeds. Air resistance introduces a damping force that causes the pendulum's motion to deviate slightly from perfect SHM, especially at larger amplitudes. For most practical purposes and small amplitudes, these effects are minimal and the simple pendulum model remains a good approximation.

Prerequisites

Understanding the periodic nature of Simple Harmonic Motion (SHM) and simple pendulums is a fascinating aspect of physics that builds upon several fundamental concepts. While there are no specific prerequisite topics provided for this subject, it's important to recognize that a strong foundation in basic physics principles is essential for grasping these more advanced concepts.

To fully appreciate the periodic nature of SHM and simple pendulums, students should have a solid understanding of Newton's laws of motion. These laws form the backbone of classical mechanics and are crucial in explaining the forces at play in oscillatory systems. Additionally, familiarity with energy conservation principles is vital, as the interplay between kinetic and potential energy is a key feature of SHM.

A good grasp of trigonometry is also beneficial when studying SHM and simple pendulums. The sinusoidal nature of these motions is best described using trigonometric functions, making this mathematical knowledge indispensable. Furthermore, basic calculus concepts, particularly differentiation and integration, are helpful in deriving equations of motion and understanding the relationships between position, velocity, and acceleration in these systems.

Students should also be comfortable with vector analysis, as it plays a role in describing the forces and motions in multiple dimensions. This is especially relevant when considering more complex pendulum systems or when analyzing SHM in different contexts.

An understanding of waves and oscillations in general provides a broader context for SHM and simple pendulums. These topics share many characteristics, and the principles learned in one area often apply to the other, reinforcing the interconnected nature of physics concepts.

Lastly, familiarity with experimental methods and data analysis is crucial. Many of the principles governing SHM and simple pendulums are best understood through hands-on experiments and careful observation. The ability to collect, analyze, and interpret data is an essential skill in physics that applies directly to these topics.

By building a strong foundation in these areas, students will be better equipped to explore the intricacies of SHM and simple pendulums. The periodic nature of these phenomena becomes more intuitive and less abstract when viewed through the lens of these prerequisite topics. As students delve deeper into the subject, they'll find that their understanding of these fundamental concepts will be reinforced and expanded, creating a rich, interconnected web of physical knowledge.