Name: simple harmonic motion: period and phase
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Ohio’s Learning Standards and Model Curriculum

9-12

Physics

Forces, Momentum And Motion

Elastic forces

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Characteristics of Simple Harmonic Motion

Considérons une règle dont une extrémité est fixée à une table et l’autre extrémité à laquelle est attachée une masse. Lorsque le système de masse dirigeante est tiré vers le haut à partir de l’extrémité libre et relâché, il exécute un mouvement harmonique simple. Si le déplacement est tracé par rapport au temps, il en résulte une forme d’onde sinusoïdale.
Étant donné que le mouvement oscillatoire commence lorsque la masse est déplacée de la position d’équilibre, l’équation du déplacement peut être exprimée comme l’amplitude, A, multipliée par la fonction cosinus de la fréquence angulaire dépendante du temps et de la phase initiale.
Le déphasage décrit la différence entre deux formes d’onde similaires en termes d’intervalle de temps et est mesuré en radians.
Rappelons que la dérivée première de la fonction de déplacement par rapport au temps est le vecteur vitesse, et la dérivée seconde est l’accélération.
Dans le mouvement harmonique simple, la vitesse est une fonction sinusoïdale qui est en retard sur le déplacement de la phase pi de 2 et qui est maximale à l’équilibre, tandis que l’accélération est une fonction cosinus qui est en retard de la phase pi et qui est maximale aux positions de déplacement maximal.

<ol>
	<li>OpenStax. (2019). Physique universitaire Vol. 1. [Version Web]. Extrait de (p. n° 745-750) <a href="https://openstax.org/books/university-physics-volume-1/pages/1-introduction">https://openstax.org/books/university-physics-volume-1/pages/1-introduction</a></li>
	<li>OpenStax. (2020). Physique au collège [Version Web]. (P. N° 671-675) <a href="https://openstax.org/details/books/college-physics">https://openstax.org/details/books/college-physics</a></li>
	<li>Halliday, Resnick et Walker (2014), Fundamentals of Physics, 10e édition, Wiley and Sons Inc. (p. n° 413-417)</li>
</ol>

caractéristiques du mouvement harmonique simple

La caractéristique clé du mouvement harmonique simple est que l'accélération du système et donc la force nette sont proportionnelles au déplacement et agissent dans la direction opposée au déplacement. De plus, la période et la fréquence d'un oscillateur harmonique simple sont indépendantes de son amplitude. Par exemple, les plongeoirs se déplacent plus vite ou plus lentement en fonction de leur épaisseur. Un plongeoir rigide et épais a une grande constante de force, ce qui lui donne une période plus petite, tandis qu'un plongeoir fin se déplace plus rapidement avec une période plus élevée. La période d'un oscillateur harmonique simple est également affectée par sa masse. Une personne plus lourde sur un plongeoir rebondit plus lentement qu'une personne avec un poids relativement plus léger.

Le mouvement harmonique simple d'un système se traduit par une onde sinusoïdale et cosinusoidale. À mesure que l'amplitude change, les maxima et les minima de la courbe changent également, mais la période et la fréquence restent inchangées. En modifiant l'angle de phase, la courbe se déplace vers la droite pour des valeurs négatives et vers la gauche pour des valeurs positives. En général, les valeurs d'angle de phase négatives décalent la courbe cosinusoidale standard vers la droite.

Caractéristiques du mouvement harmonique simple

La caractéristique clé du mouvement harmonique simple est que l’accélération du système et, par conséquent, la force résultante sont proportionnelles à la ...

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Unit 1

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Oscillations

KEY TERMS AND CONCEPTS

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simple harmonic motion: period and phase

Simple harmonic motion depends on how displacement, force, and acceleration work together. In this motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and points in the opposite direction. That restoring force pulls the object back toward equilibrium.

The period and frequency of a simple harmonic oscillator do not depend on amplitude. A diving board shows this idea well. A stiff, thick diving board has a large force constant, so it has a smaller period. A thin board moves differently and has a higher period.

Mass also affects the period of simple harmonic motion. A heavier person on a diving board bounces up and down more slowly than a lighter person. The motion of the system follows a sine and cosine waveform. When amplitude changes, the maxima and minima of the curve change, but the period and frequency stay the same.

Phase angle changes the position of the wave on the graph. Negative phase angle values shift the standard cosine curve to the right, while positive values shift it to the left. In general, a negative phase angle moves the cosine curve rightward.

Simple Harmonic Motion: Period and Phase

simple harmonic motion

Period and Frequency in Simple Harmonic Motion

Simple harmonic motion describes oscillatory motion when the net force follows Hooke's law. In this type of motion, the force acts toward the equilibrium position. If there is no damping from friction or other non-conservative forces, the oscillator moves with equal displacement on both sides of equilibrium. The period of a simple harmonic oscillator is found from the equation of motion. Period is the time for one complete oscillation. Frequency is the inverse of period, so the two values can...

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characteristics of simple harmonic motion

Simple harmonic motion depends on how displacement, force, and acceleration work together. In this motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and points in the opposite direction. That restoring force pulls the object back toward equilibrium. The period and frequency of a simple harmonic oscillator do not depend on amplitude. A diving board shows this idea well. A stiff, thick diving board has a large force constant, so it has a...

oscillations about an equilibrium position

oscillations-about-an-equilibrium-position

Stable Equilibrium and Atomic Oscillation

Stable equilibrium is a key idea in oscillation. When an object sits at a stable equilibrium point, a small push moves it away, but it then tends to oscillate around that point. If the equilibrium point is unstable, a slight disturbance will not bring the object back. The transcript also describes a half-stable equilibrium point. It is still considered unstable, but the name comes from the fact that the force on only one side points toward the equilibrium point. That makes it different from a...

energy in simple harmonic motion

Simple Harmonic Motion: Energy and Speed

Simple harmonic motion combines potential energy and kinetic energy. In a spring-like oscillator, Hooke’s law explains the potential energy stored when the string or spring is stretched or compressed. Because there are no dissipative forces, the object also has kinetic energy as it moves.

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frequency of spring-mass system

How Mass and Spring Stiffness Set Oscillation

A spring-mass system shows simple harmonic motion, and its oscillation rate depends on the mass and the spring force constant. In this model, the angular frequency, period, and frequency are set by those two values. They do not depend on the motion’s amplitude or on the initial conditions. To find these relationships, the system can be analyzed with Newton’s second law. Imagine a block on a spring on a frictionless surface. Three forces act on the mass: weight, the normal force, and the spring...

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simple harmonic motion and uniform circular motion

simple-harmonic-motion-and-uniform-circular-motion

Simple Harmonic Motion from Circular Motion

Simple harmonic motion and uniform circular motion are closely linked. Simple harmonic motion is oscillatory motion in a system where the net force follows Hooke’s law. Uniform circular motion is motion along a circular path at constant speed. A simple way to create simple harmonic motion is to use uniform circular motion. One example is a ball attached to a vertical turntable that spins at a steady rate. If you watch the ball’s shadow on the floor, the shadow, or projection, moves in simple...

problem solving: energy in simple harmonic motion

problem-solving-energy-in-simple-harmonic-motion

Energy Conservation in Simple Harmonic Motion

Energy conservation in simple harmonic motion explains how a spring and mass system shares energy as it moves back and forth around equilibrium. Simple harmonic motion, or SHM, is periodic motion in time and position. The object oscillates with a constant amplitude and frequency. In SHM, kinetic energy and potential energy change into each other continuously. A good example is the spring in a car shock absorber as the car moves on a bumpy road. The wheel attached to the spring oscillates in...

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simple pendulum

Pendulum Motion and Period

A simple pendulum is a small ball, or bob, hanging from a string that has negligible mass and does not stretch. Pendulums appear in everyday life, including clocks, swing sets, and a sinker on a fishing line. Their motion is a useful model for studying repeating movement. The period of a simple pendulum depends on two factors: its length and the acceleration due to gravity. The period does not depend on the bob’s mass or on the maximum displacement. For small angles, the pendulum behaves like...

torsional pendulum

Angular Oscillation in a Torsion String

Angular oscillation in a torsion string is shown by a rigid body suspended from a string that twists as it moves. The twisting of the string provides the restoring force. The string is ideally massless, and in practice its mass is much smaller than the rigid body&amp;#39;s mass, so it is neglected. When the angular displacement is small, the motion can be modeled as linear angular oscillation. The amplitude is measured as an angle. In this system, the role of mass is played by the rigid...

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physical pendulum

Physical Pendulum Motion and Inertia

A physical pendulum is a rigid body that hangs from a fixed pivot point and swings when it is displaced. Its motion is similar to a simple pendulum. To describe that motion, physicists use the small-angle approximation, where sin θ ≈ θ. This approximation is valid up to about 14°. The period and angular frequency of a physical pendulum can be found by comparing it with a spring-mass system. The mass moment of inertia is the key quantity that describes how the body’s mass is spread around the...

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measuring acceleration due to gravity

Using Pendulums to Find Gravity

Pendulums can be used to measure the acceleration due to gravity. A simple pendulum is a point mass attached to a string. A physical pendulum is an object that swings in a similar way, but its mass is spread out, so it cannot be treated as a point mass on a string. A physical pendulum is described using rotational motion and moment of inertia. In both types of pendulum, gravity provides the restoring force that brings the object back toward the middle of its swing. For a simple pendulum,...

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damped oscillations

Energy Loss in Oscillations

Oscillations in the real world often lose energy as they move. A system that keeps the same amplitude without any loss is called undamped. That kind of motion is rare, because friction and other non-conservative forces usually reduce the motion over time. Common examples show this clearly. A guitar string stops vibrating a few seconds after it is plucked. A swing also needs a person to keep pushing it if the motion is to continue. In some devices, such as car shock absorbers, damping is useful...

types of damping

How Damping Changes Oscillations

Damping changes how a system oscillates by reducing its back-and-forth motion. As damping increases, the period and frequency begin to change because damping opposes motion and lowers the net force in both directions. When damping is very large, the system stops oscillating altogether. It then moves slowly toward equilibrium. An overdamped system reaches equilibrium slowly. An underdamped system reaches equilibrium faster, but it oscillates around the equilibrium point while doing so. A...

forced oscillations

Resonance in Driven Oscillators

Resonance in driven oscillators describes how a periodic force changes an object’s motion. A driven oscillator may first move in a messy way. These early motions are called transients. After the transients die out, the oscillator reaches a steady state. In this state, the motion becomes periodic and the displacement is determined. The amplitude of a driven oscillator can be found from the equations for force and motion. That amplitude depends on the natural frequency of the system and the...

concept of resonance and its characteristics

concept-of-resonance-and-its-characteristics

Resonance in Music, Cars, and Radios

Resonance is a key feature of driven oscillators, and it depends strongly on damping. Damping is the process that reduces motion by removing energy from the system. When a system must resonate at one specific frequency, very light damping is needed. Piano strings and many other musical instruments are common examples of this kind of response. Heavy damping works better when the goal is small-amplitude oscillations. A car suspension system uses this approach so that bumps do not produce large...

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key terms and concepts

Explore how simple harmonic motion links displacement, force, period, mass, amplitude, and phase angle, with examples from a diving board and wave graphs.

Simple Harmonic Motion: Period and Phase | JoVE K12

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PHYS.8

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Force

Forces affect every moment of our life. Our bodies are held to the Earth by force, and they are held together by the forces of charged particles. When we ...

Types of Forces

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Newton's First Law: Introduction

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Newton's First Law: Application

Experience suggests that an object at rest remains at rest if left alone, and that an object in motion tends to slow down and stop unless some effort is ...

Internal and External Forces

Newton&#39;s first law states that a net external force causes a change in motion. External forces act on an object or system, originating outside of the ...

Newton's Second Law

Newton&#39;s second law is closely related to his first law of motion. It mathematically gives the cause-and-effect relationship between force and changes ...

Mass and Weight

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Newton's Third Law: Introduction

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Newton's Third Law: Examples

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Second Law: Motion under Same Force

Newton&#39;s laws can be applied to bodies at rest and bodies in motion. Newton&#39;s first law is applied to bodies in equilibrium, whereas the second ...

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Second Law: Motion under Same Acceleration

Newton&#39;s second law of motion applies to bodies moving under the same acceleration. For example, when a baggage tractor pulls luggage carts, each cart ...

Frictional Force

When a body is in motion, it encounters resistance because the body interacts with its surroundings. This resistance is known as friction, a common yet ...

Static and Kinetic Frictional Force

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Frequency of Spring-Mass System

One interesting characteristic of the simple harmonic motion (SHM) of an object attached to a spring is that the angular frequency, and the period and ...

Newton's laws applied to complex problems

Content elaboration: Newton's laws of motion, especially the third law, can be used to solve complex problems that involve systems of many objects that move together as one (e.g., an Atwood machine). The equation $a=F_{net}/m$ that was introduced in physical science can be used to solve more complex problems involving systems of objects and situations involving forces that must themselves be quantified (e.g., gravitational forces, elastic forces, friction forces).

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Newton's laws applied to complex problems

Gravitation

In the years before Newton, a general belief prevailed that different laws governed objects in the sky than objects on Earth. When Kepler wrote down the ...

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Newton's Law of Gravitation

Our everyday observation tells us that all objects close to the Earth naturally tend to fall to the ground. Early philosophers assumed that this downward ...

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Gravitation Between Spherically Symmetric Masses

The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, ...

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Gravity between Spherical Bodies

Newton&#39;s law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, ...

Apparent Weight

True weight is the measure of the gravitational force acting on an object. However, if the object accelerates, its measured weight is different from its ...

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Apparent Weight and the Earth's Rotation

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The Principle of Superposition and the Gravitational Field

The principle of superposition applies to gravitational forces of objects that are sufficiently far apart. It states that the net gravitational force on a ...

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Gravitational force and fields

Content elaboration: Gravitational interactions are very weak compared to other interactions and are difficult to observe unless one of the objects is extremely massive (e.g., the sun, planets, moons). The force law for gravitational interaction states that the strength of the gravitational force is proportional to the product of the two masses and inversely proportional to the square of the distance between the centers of the masses, $F_{g}=(G\cdot m_{1}\cdot m_{2})/r^{2}$. The proportionality constant, G, is called the universal gravitational constant and has a value of $6.674\cdot10^{-11}m^{3}/(kg\cdot s^{2})$. Problem solving may involve calculating the net force for an object between two massive objects (e.g., Earth-moon system, planet-sun system) or calculating the position of such an object given the net force. The strength of an object's (i.e., the source's) gravitational field at a certain location, g, is given by the gravitational force per unit of mass experienced by another object placed at that location, $g=F_{g}/m$. Comparing this equation to Newton's second law can be used to explain why all objects on Earth's surface accelerate at the same rate in the absence of air resistance. While the gravitational force from another object can be used to determine the field strength at a particular location, the field of the object is always there, even if the object is not interacting with anything else. The field direction is toward the center of the source. Given the gravitational field strength at a certain location, the gravitational force between the source of that field and any object at that location can be calculated. Greater gravitational field strengths result in larger gravitational forces on masses placed in the field. Gravitational fields can be represented by field diagrams obtained by plotting field arrows at a series of locations. Field line diagrams are excluded from this course. Distinctions between gravitational and inertial masses are excluded. A scale indicates weight by measuring the normal force between the object and the surface supporting it. The reading on the scale accurately measures the weight if the system is not accelerating. However, if the scale is used in an accelerating system, as in an elevator, the reading on the scale does not equal the actual weight. The scale reading can be referred to as the "apparent weight." This apparent weight in accelerating elevators can be explained and calculated using force diagrams and Newton's laws.

Gravitational force and fields

Simple Harmonic Motion

Simple harmonic motion is the name given to oscillatory motion for a system where the net force can be described by Hooke&#39;s law. If the net force can ...

The key characteristic of the simple harmonic motion is that the acceleration of the system and, therefore, the net force are proportional to the ...

Oscillations about an Equilibrium Position

Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable ...

Hooke's Law

Hooke&#39;s law states that stress applied to a material is directly proportional to the strain it experiences within the elastic limit. For ductile ...

Stress

When a force is applied on a body, it undergoes deformation. In order to restore the body to its original shape and/or size, an opposite or restoring ...

Strain and Elastic Modulus

The quantity that describes the deformation of a body under stress is known as strain. Strain is given as a fractional change in either length, volume, or ...

Elasticity

Elasticity is the ability of an object to withstand the effects of distortion and to return to its original size and shape once the forces causing ...

Elastic forces

Content elaboration: Elastic materials stretch or compress in proportion to the load they support. The mathematical model for the force that a linearly elastic object exerts on another object is $F_{elastic}=k\Delta x$ where $\Delta x$ is the displacement of the object from its relaxed position. The direction of the elastic force is always toward the relaxed position of the elastic object. The constant of proportionality, k, is the same for compression and extension and depends on the "stiffness" of the elastic object.

Friction force (static and kinetic)

Content elaboration: The amount of kinetic friction between two objects depends on the electric forces between the atoms of the two surfaces sliding past each other. It also depends upon the magnitude of the normal force that pushes the two surfaces together. This can be represented mathematically as $F_{k}=\mu_{k}F_{N}$, where $\mu_{k}$ is the coefficient of kinetic friction that depends upon the materials of which the two surfaces are made. Sometimes friction forces can prevent objects from sliding past each other, even when an external force is applied parallel to the two surfaces that are in contact. This is called static friction, which is mathematically represented by $F_{s}\le\mu_{s}F_{N}$ The maximum amount of static friction possible depends on the types of materials that make up the two surfaces and the magnitude of the normal force pushing the objects together, $F_{smax}=\mu_{s}F_{N}$ As long as the external net force is less than or equal to the maximum force of static friction, the objects will not move relative to one another. In this case, the actual static friction force acting on the object will be equal to the net external force acting on the object, but in the opposite direction. If the external net force exceeds the maximum static friction force for the object, the objects will move relative to each other and the friction between them will no longer be static friction, but will be kinetic friction.

Friction force (static and kinetic)

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Density

Density is an important characteristic of substances, crucial in determining whether an object sinks or floats in a fluid. Its SI unit is kg/m3, and its ...

Viscosity

When water is poured into a glass, it falls freely and quickly, whereas if honey or maple syrup is poured over a pancake, it flows slowly and sticks to ...

Air resistance and drag

Content elaboration: Liquids have more drag than gases. When an object pushes on the particles in a fluid, the fluid particles can push back on the object according to Newton's third law and cause a change in motion of the object. This is how helicopters experience lift and how swimmers propel themselves forward. Forces from fluids are quantified using Newton's second law and force diagrams. Factors that affect air resistance and drag and the determination of terminal velocity may be included.

Air resistance and drag

Introduction to Vectors

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Vector Algebra: Graphical Method

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Principle of Linear Impulse and Momentum for a Single Particle: Problem Solving

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Forces in two dimensions
• Adding vector forces
• Motion down inclines
• Centripetal forces and circular motion

Content elaboration: Net forces will be calculated for force vectors with directions between $0^{\circ}$ and 360° or a certain angle from a reference (e.g., 37° above the horizontal). Vector addition can be done with trigonometry or by drawing scaled diagrams. Problems can be solved for objects sliding down inclines. The net force, final velocity, time, displacement and acceleration can be calculated. Inclines will either be frictionless or the force of friction will already be quantified. Calculations of friction forces down inclines from the coefficients of friction and the normal force will not be addressed in this course. An object moves at constant speed in a circular path when there is a constant net force that is always directed at right angles to the direction of motion toward the center of the circle. In this case, the net force causes an acceleration that shows up as a change in direction. If the force is removed, the object will continue in a straight-line path. The nearly circular orbits of planets and satellites result from the force of gravity. Centripetal acceleration is directed toward the center of the circle and can be calculated by the equation $ac=v^{2}/r$, where v is the speed of the object and r is the radius of the circle. This expression for acceleration can be substituted into Newton's second law to calculate the centripetal force. Since the centripetal force is a net force, it can be equated to friction (unbanked curves), gravity, elastic force, etc., to perform more complex calculations.

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Linear Momentum

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Content elaboration: Momentum, p, is a vector quantity that is directly proportional to the mass, m, and the velocity, v, of the object. Momentum is in the same direction the object is moving and can be mathematically represented by the equation $p=mv$ The conservation of linear momentum states that the total (net) momentum before an interaction in a closed system is equal to the total momentum after the interaction. In a closed system, linear momentum is always conserved for elastic, inelastic and totally inelastic collisions. While total energy is conserved for any collision, in an elastic collision, the kinetic energy also is conserved. Given the initial motions of two objects, qualitative predictions about the change in motion of the objects due to a collision can be made. Problems can be solved for the initial or final velocities of objects involved in inelastic and totally inelastic collisions. Momentum may be dealt with in two dimensions conceptually, but at this level calculations should be limited to only one dimension. Coefficients of restitution are beyond the scope of this course. Impulse, $\Delta p$ is the total momentum transfer into or out of a system. Any momentum transfer is the result of interactions with objects outside the system and is directly proportional to both the average net external force acting on the system, $F_{avg}$, and the time interval of the interaction, t. It can mathematically be represented by $\Delta p=p_{f}-p_{i}=F_{avg}\Delta t$ This equation can be used to justify why momentum changes due to the external force of friction can be ignored when the time of interaction is extremely short. Average force, initial or final velocity, mass or time interval can be calculated in multi-step word problems. For objects that experience a given impulse (e.g., a truck coming to a stop), a variety of force/time combinations are possible. The time could be small, which would require a large force (e.g., the truck crashing into a brick wall to a sudden stop). Conversely, the time could be extended which would result in a much smaller force (e.g., the truck applying the brakes for a long period of time).

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Momentum, impulse and conservation of momentum

Simple Harmonic Motion: Period and Phase

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