Periodic Motion

Have you ever wondered how a guitar works and produces a mesmerizing sound wherein when you pluck the strings it will automatically move back and forth and you can also feel that the guitar vibrates. Well, to explain that, the guitar strings undergo periodic motion when they vibrate. When a guitar string is plucked, it is displaced from its equilibrium position and then released. The tension in the string, along with its elasticity, causes it to return to its original position. However, due to its inertia, it overshoots and gets displaced in the opposite direction. This back-and-forth motion continues, resulting in the string vibrating at its natural frequency. The vibrating guitar string produces sound waves, which we perceive as sound when they reach our ears. The pitch or frequency of the sound depends on the frequency of the string's vibration, which is determined by factors such as the tension in the string, its length, and its mass per unit length. By pressing down on different frets of the guitar neck, the effective length of the vibrating portion of the string can be changed, altering its frequency and producing different musical notes. Periodic motion is applied in different ways but it is the movement thoroughly repeated in equal intervals of time (The Editors of Encyclopaedia Britannica, 1998).

Motivation(Find Relevant Words)

Motivation

Answersheet:
FREQUENCY
DISPLACEMENT
PERIOD
AMPLITUDE
ANGULAR
VELOCITY
MOTION

Objectives

Relate the amplitude, frequency, angular frequency, period, displacement, velocity, and acceleration of oscillating systems
Recognize the necessary conditions for an object to undergo simple harmonic motion
Analyze the motion of an oscillating system using energy and Newton’s 2nd law approaches
Calculate the period and the frequency of spring mass, simple pendulum, and physical pendulum
Differentiate underdamped, overdamped, and critically damped motion
Describe the conditions for resonance

Displacement (x):

This is the distance an oscillating object is from its equilibrium position (rest position) at a specific time, denoted by x(t). Positive displacement indicates a movement away from equilibrium in one direction, and negative displacement signifies movement in the opposite direction. Amplitude (explained below) defines the maximum displacement from equilibrium

Amplitude (A):

This signifies the maximum absolute value of the displacement, representing how far the object moves away from its equilibrium position during oscillation. A larger amplitude indicates a more vigorous oscillation with higher energy.

Period (T):

This represents the time it takes for one complete cycle of oscillation. In other words, it's the time taken for the object to move from its equilibrium position, reach the maximum displacement in one direction, return to equilibrium, reach the maximum displacement in the opposite direction, and finally come back to the equilibrium position again.

Frequency (f):

This refers to the number of oscillation cycles completed per unit time (usually seconds). It's the inverse of the period (f = 1/T). A higher frequency signifies more frequent oscillations, meaning the object completes more cycles in a shorter time.

Angular Frequency (ω):

Closely related to frequency, angular frequency is measured in radians per second (rad/s). It relates the rate of change of the oscillating object's phase angle (which tracks the position within the cycle) to time. The two are connected by a constant value (ω = 2πf). A higher angular frequency indicates a faster rotation of the phase angle and consequently, a higher frequency of oscillation.

Velocity (v):

This describes how fast the oscillating object's position changes with time. It's the rate of change of displacement (v(t) = dx(t)/dt). Velocity is positive when the object moves away from equilibrium and negative when it moves towards equilibrium. The velocity is zero when the object is momentarily at its equilibrium position and reaches its maximum values when the object is farthest from equilibrium (at the maximum displacement).

Acceleration (a):

This signifies how quickly the oscillating object's velocity changes with time. It's the rate of change of velocity (a(t) = dv(t)/dt). Acceleration is positive when the object moves towards its equilibrium position (restoring force acting) and negative when it moves away from equilibrium. The acceleration is zero when the object passes through its equilibrium position.

Simple harmonic motion (SHM) is oscillatory motion for a system where the restoring force is proportional to the displacement and acts in the direction opposite to the displacement.
In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and acts in the opposite direction of the displacement.
The restoring force must be proportional to the displacement and act opposite to the direction of motion with no drag forces or friction. The frequency of oscillation does not depend on the amplitude.
What is so significant about SHM? For one thing, the period T and frequency f of a simple harmonic oscillator are independent of amplitude.
Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant (k), which causes the system to have a smaller period. For example, you can adjust a diving board’s stiffness—the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one. In fact, the mass m and the force constant k are the only factors that affect the period and frequency of SHM. To derive an equation for the period and the frequency, we must first define and analyze the equations of motion. Note that the force constant is sometimes referred to as the spring constant.
One interesting characteristic of the SHM of an object attached to a spring is that the angular frequency, and therefore the period and frequency of the motion, depend on only the mass and the force constant, and not on other factors such as the amplitude of the motion. We can use the equations of motion and Newton’s second law (→Fnet=m→a) to find equations for the angular frequency, frequency, and period.

What is a simple harmonic motion?

- Simple harmonic motion (SHM) is a specific type of periodic motion. Simple harmonic motion is a specific type of periodic motion characterized by a restoring force that is directly proportional to the displacement of the object from its equilibrium position and acts in the opposite direction. This results in a motion that is sinusoidal, oscillating back and forth around an equilibrium position. Examples of systems exhibiting SHM include the motion of a mass-spring system, the oscillation of a pendulum under small angles, and the vibrations of atoms in a lattice.

- Simple harmonic motion is a physics concept that describes a repetitive oscillation around an equilibrium point, where the maximum displacement from this point is equal on both sides. Each full cycle of this oscillation takes the same amount of time. The force driving this motion always acts towards the equilibrium position and increases or decreases in proportion to the distance from it.

How can we say an object is in simple harmonic motion?

There should be no energy dissipation in the system. This means that there should be no friction or other forms of damping present that would cause the energy of the system to decrease over time.

The restoring force must act linearly, meaning that the force applied is directly proportional to the displacement.

The period of oscillation remains unchanged regardless of the amplitude (maximum displacement) of the motion.

:The frequency of oscillation, or the number of cycles per unit of time, remains constant throughout the motion.

Application In Real Life

Pendulum Clock

The swinging motion of a pendulum in a clock exhibits simple harmonic motion. The period of the pendulum's oscillation remains constant, allowing accurate timekeeping.

Sample Problems:

Calculate the length of string needed for a simple pendulum if the period is 2.0 s and the mass is 2.0 kg.
A simple pendulum consisting of a mass of 10.0 kg is pulled 8.0 cm from the vertical and released. If the string the mass is suspended from is 1.5 m long, calculate the maximum velocity of the mass.

Resonance

Resonance is a phenomenon that can enhance periodic motion by adding energy to a system at its natural frequency. It occurs when an external force matches the natural frequency of the system, leading to amplification of oscillations.

Summary

The "Harmonic Oscillator Challenge" is a game where players navigate through levels representing different oscillating systems. They control parameters like stiffness, mass, and length to achieve specific oscillation characteristics such as frequency, period, and amplitude. Every level focuses on a particular system, such as a physical pendulum, a spring-mass arrangement, or a basic pendulum, enabling players to comprehend how these elements affect oscillation behavior. Along with learning about underdamped, overdamped, critically damped motion, and resonance states, players also confront damping scenarios and resonance problems. The goal of the game is to provide an immersive and participatory experience while helping players gain a deeper grasp of the dynamics and features of oscillating systems.

Periodic motion encompasses a wide array of phenomena in physics, characterized by repeating patterns over time. One fundamental form is simple harmonic motion (SHM), where an object oscillates back and forth around an equilibrium position under the influence of a restoring force proportional to displacement. Key elements of SHM include displacement, amplitude (the maximum displacement), period (the time for one complete cycle), frequency (the number of cycles per unit time), angular frequency (related to frequency by a constant factor), velocity, and acceleration. These parameters govern the behavior of oscillatory systems and are crucial in understanding their dynamics.

Real-world examples of periodic motion include the swinging of a pendulum in a clock, the vibrations of guitar strings when plucked, and the oscillations of atoms in a lattice. Periodic motion is also prevalent in electromagnetic waves, such as radio waves, which exhibit sinusoidal oscillations in electric and magnetic fields. Resonance is another important concept related to periodic motion, where external forces can amplify oscillations by matching the natural frequency of a system. Understanding damping scenario such as overdamping, underdamping, and critically damped motion, they are vital for predicting and controlling the behavior of oscillatory systems in various engineering and scientific applications. In essence, periodic motion is present everywhere in nature and plays a fundamental role in understanding and describing the dynamics of physical systems across different scales.