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Periodic Motion: Springs and Pendulums

In this article, you will learn more about periodic motion, especially springs and pendulums in order to provide the best exam preparation.

Last updated: Mar 8, 2022

Editorial responsibility: Stanley Oiseth, Lindsay Jones, Evelin Maza

Springs: Definition

Springs are elastic Elastic Connective Tissue: Histology objects used to store mechanical energy in their coils owing to their ability to stretch and/or be compressed. They are usually referred to as coil springs. As a spring is compressed from its resting (non-mobile) position, it has the capability of exerting an opposing force in relation to the change in length it has undergone.

Restorative force capability

Mechanical energy exists as kinetic and potential energies; the former is present when the spring is unstretched, while the latter is the energy in the system when the spring is stretched. When kinetic energy increases, the elastic Elastic Connective Tissue: Histology potential energy correspondingly decreases.

The opposing force existing in the mechanical aspects of the spring is referred to as restorative force and is based on Hooke’s Law. This is the force required to restore the spring to its resting or equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy position.

The restorative force is dependent on the spring constant and displacement Displacement The process by which an emotional or behavioral response that is appropriate for one situation appears in another situation for which it is inappropriate. Defense Mechanisms of the spring from its equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy position. The spring constant in turn is dependent on the strength of the spring and takes into consideration various physical properties such as the material of the spring and the diameter of the spring coil. The thicker the spring, the higher the spring constant. Conversely, thinner springs have lower spring constants.

The equation for restorative force can be represented as follows:

Fspring = – k x

k = spring constant

x = displacement Displacement The process by which an emotional or behavioral response that is appropriate for one situation appears in another situation for which it is inappropriate. Defense Mechanisms of the spring from the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy position

Fspring = restorative force

The equation requires a negative sign since the spring experiences a force pulling it back to its equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy position upon being stretched or compressed. This force acts in the opposite direction of the initial stretch or compression Compression Blunt Chest Trauma. In other words, the direction in which the spring stretches is opposite to the direction of the force it exerts.

The spring constant is always positive since only the magnitude is considered. It is represented as Newton per centimeter (N/cm) or Newton per millimeter (N/ mm MM Multiple myeloma (MM) is a malignant condition of plasma cells (activated B lymphocytes) primarily seen in the elderly. Monoclonal proliferation of plasma cells results in cytokine-driven osteoclastic activity and excessive secretion of IgG antibodies. Multiple Myeloma).

Springs have an elastic Elastic Connective Tissue: Histology limit Limit A value (e.g., pressure or time) that should not be exceeded and which is specified by the operator to protect the lung Invasive Mechanical Ventilation, which refers to the extent to which it can be stretched without altering its ability to return to its original length. As long as this limit Limit A value (e.g., pressure or time) that should not be exceeded and which is specified by the operator to protect the lung Invasive Mechanical Ventilation is not exceeded, the spring displays elastic Elastic Connective Tissue: Histology behavior; however, if exceeded, it portrays plastic behavior. Hooke’s law holds true only if the spring is within its elastic Elastic Connective Tissue: Histology limit Limit A value (e.g., pressure or time) that should not be exceeded and which is specified by the operator to protect the lung Invasive Mechanical Ventilation.

Springs have restorative forces

Image by Lecturio.

Oscillating ability

Springs can oscillate; in other words, they can produce a repetitive variation in motion between their two extreme states, stretched and compressed, starting from a midpoint (the resting or equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point).

When a spring is stretched and held in place, an object at the end of the spring has no velocity since it is not in motion. The force will act in a direction opposite to that of the stretch. This explains the negative sign in the equation representing Hooke’s Law. This force tries to restore the position of the spring to its equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point. The displacement Displacement The process by which an emotional or behavioral response that is appropriate for one situation appears in another situation for which it is inappropriate. Defense Mechanisms is denoted as ‘x,’ while the maximum displacement Displacement The process by which an emotional or behavioral response that is appropriate for one situation appears in another situation for which it is inappropriate. Defense Mechanisms that the spring can undergo is represented as the maximum amplitude (A). When an object at the end of the spring is released, it will be pulled towards the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point with velocity ‘v.’

At the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point, ‘x’ would equal 0, since the spring is passing through this point as it is being compressed to oppose the stretch. The spring continues to compress owing to its inherent velocity and passes equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy until it reaches the minimum amplitude at which the velocity is 0. At this point, the spring possesses a restorative force that acts in the opposite direction enabling it to return to equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy.

This cycle Cycle The type of signal that ends the inspiratory phase delivered by the ventilator Invasive Mechanical Ventilation continues until the spring finally stops at the resting point. This back and forth motion comprising stretch and compress is termed an oscillation. Upon completion of several oscillations, the spring stops moving and returns to its resting point, where both ‘x’ and ‘v’ equal 0.

Spring passes equilibrium

Image by Lecturio.

Spring period and frequency calculations

Various aspects can be determined based on spring oscillations and their properties. The frequency ‘f’ indicates the number of cycles per second the spring undergoes, while the period ‘P’ denotes the time between oscillating motions. The two traits bear an inverse relationship Relationship A connection, association, or involvement between 2 or more parties. Clinician–Patient Relationship and can be represented as P = 1/f.

Let us look at the frequency to determine the properties that affect Affect The feeling-tone accompaniment of an idea or mental representation. It is the most direct psychic derivative of instinct and the psychic representative of the various bodily changes by means of which instincts manifest themselves. Psychiatric Assessment this variable Variable Variables represent information about something that can change. The design of the measurement scales, or of the methods for obtaining information, will determine the data gathered and the characteristics of that data. As a result, a variable can be qualitative or quantitative, and may be further classified into subgroups. Types of Variables. The frequency is affected by the spring constant ‘k’ and mass Mass Three-dimensional lesion that occupies a space within the breast Imaging of the Breast of the spring ‘m.’

f = (1/2 π) √(k/m)

Thus, P = 2 π √(m/k).

Energy conservation of a spring

Let us now look at how kinetic and potential energies affect Affect The feeling-tone accompaniment of an idea or mental representation. It is the most direct psychic derivative of instinct and the psychic representative of the various bodily changes by means of which instincts manifest themselves. Psychiatric Assessment oscillations. When a spring is stretched and subsequently released, it moves through the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point. The initial energy prior to release Release Release of a virus from the host cell following virus assembly and maturation. Egress can occur by host cell lysis, exocytosis, or budding through the plasma membrane. Virology equals the energy as the spring moves through the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point. This is the law of conservation of energy in a spring.

Thus,

Einitial = Efinal

Energy is a combination of kinetic energy ‘K’ and potential energy ‘U.’

So,

Kinitial + Uinitial = Kfinal + Ufinal

Before the spring is released after being stretched, the system can be considered to be at rest. At this point, kinetic energy does not come into play. Thus, Kinitial = 0. Similarly, since there is no potential energy at the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy state, Ufinal = 0. This simplifies the equation to:

Uinitial = Kfinal

(1/2) k A2 = (1/2) m v2

This equation can be expanded, since potential energy is dependent on the stretched spring, whereas kinetic energy is dependent on the velocity of the spring motion.

Solving for velocity:

v = A

In other words, a stretched spring possesses only potential energy. Upon release Release Release of a virus from the host cell following virus assembly and maturation. Egress can occur by host cell lysis, exocytosis, or budding through the plasma membrane. Virology, the restorative forces convert potential energy to kinetic energy and subsequently, at the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy position, the spring possesses only kinetic energy. Conversely, when the spring compresses back to its minimal amplitude, its kinetic energy is completely converted to potential energy. This is similar to the potential- and kinetic-energy changes that occur due to the effect of gravity on an object.

Energy conservation in springs

Image by Lecturio.

Applications

This principle is used in the manufacture of kitchen scales Scales Dry or greasy masses of keratin that represent thickened stratum corneum. Secondary Skin Lesions as well as devices that use springs, such as the suspension system in cars and retractable pens.

Pendulums: Definition

A pendulum is a weight suspended from a frictionless pivot enabling it to swing freely. Initially, before swinging, the pendulum rests straight downward at its resting or equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy position. Once it is displaced sideways, the swinging commences with oscillating back and forth motions. The pendulum swings in this manner since there is a restorative force to propel it to the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy position.

Restorative force capability

The restorative force of a pendulum drives it back to its equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point after it has been displaced. The pendulum has a restorative force and oscillating pattern similar to that of a spring. A swinging pendulum passes through the resting position and continues to swing until it reaches a certain distance past the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point; it stops here momentarily and swings back in the opposite direction.

This oscillating motion continues until the pendulum eventually stops swinging.

It is assumed that there is no friction or air resistance Resistance Physiologically, the opposition to flow of air caused by the forces of friction. As a part of pulmonary function testing, it is the ratio of driving pressure to the rate of air flow. Ventilation: Mechanics of Breathing in this model. Unlike a spring, the restorative force is dependent on gravity and the angle (ɵ) of motion from the midpoint, rather than the mass Mass Three-dimensional lesion that occupies a space within the breast Imaging of the Breast suspended from the pendulum. The pendulum is like a spring since the restorative forces for both depend on displacement Displacement The process by which an emotional or behavioral response that is appropriate for one situation appears in another situation for which it is inappropriate. Defense Mechanisms.

F = m g sin ɵ

F = restorative force to return to equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy

M = mass Mass Three-dimensional lesion that occupies a space within the breast Imaging of the Breast of the swinging object

g = gravity

ɵ = swing angle from the midpoint

However, for mathematical calculations, sin ɵ = ɵ for very small angles causing:

F = m g ɵ

Pendulum

Image by Lecturio.

Pendulum period and frequency calculations

Various aspects can be determined based on the oscillations of a pendulum. The frequency ‘f’ indicates the number of oscillations of the pendulum per second, while the period ‘P’ denotes the time between oscillating motions. The two traits are inversely related and can be represented as P = 1/f

Let us look at the frequency to determine the properties that affect Affect The feeling-tone accompaniment of an idea or mental representation. It is the most direct psychic derivative of instinct and the psychic representative of the various bodily changes by means of which instincts manifest themselves. Psychiatric Assessment this variable Variable Variables represent information about something that can change. The design of the measurement scales, or of the methods for obtaining information, will determine the data gathered and the characteristics of that data. As a result, a variable can be qualitative or quantitative, and may be further classified into subgroups. Types of Variables. The frequency is affected by the gravitational energy ‘g’ and the length of the pendulum ‘L.’

f = 1/ [ 2π  √(L/g)]

Thus,

P = 2 π √(L/g)

It is to be noted that the mass Mass Three-dimensional lesion that occupies a space within the breast Imaging of the Breast on the pendulum does not affect Affect The feeling-tone accompaniment of an idea or mental representation. It is the most direct psychic derivative of instinct and the psychic representative of the various bodily changes by means of which instincts manifest themselves. Psychiatric Assessment its swing.

Energy conservation

Pendulum oscillations are governed by kinetic and potential energies. When a pendulum is displaced and subsequently released, it moves through the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point.

The energy of the system before displacement Displacement The process by which an emotional or behavioral response that is appropriate for one situation appears in another situation for which it is inappropriate. Defense Mechanisms equals the energy possessed by the pendulum as it moves through the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point. This is the law of conservation of energy in a pendulum.

Thus, Einitial = Efinal. Since there is a difference in the vertical location of the pendulum at its initial and final stops, the equation can be depicted as

Etop = Ebottom

Etop represents the energy prior to the release Release Release of a virus from the host cell following virus assembly and maturation. Egress can occur by host cell lysis, exocytosis, or budding through the plasma membrane. Virology of the pendulum from its displaced position, while Ebottom represents the energy at the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy point.

Energy is a combination of kinetic energy ‘K’ and potential energy ‘U.’

So,

Ktop + Utop = Kbottom + Ubottom

Before being released from its initial position, the kinetic energy of the pendulum, Ktop = 0. Additionally, at the equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy position, the potential energy of the system, Ubottom = 0. This simplifies the equation to:

Utop = Kbottom

mgh = (1/2) m v2

The equation can be expanded since potential energy is dependent on the displaced pendulum, whereas kinetic energy is dependent on the pendulum velocity.

Solving for velocity:

v =  √(2gh)

In other words, a pendulum before being displaced possesses only potential energy. Upon release Release Release of a virus from the host cell following virus assembly and maturation. Egress can occur by host cell lysis, exocytosis, or budding through the plasma membrane. Virology, the restorative forces convert potential energy to kinetic energy, and subsequently, the pendulum at equilibrium Equilibrium Occurs when tumor cells survive the initial elimination attempt These cells are not able to progress, being maintained in a state of dormancy by the adaptive immune system. In this phase, tumor immunogenicity is edited, where T cells keep selectively attacking highly immunogenic tumor cells.This attack leaves other cells with less immunogenicity to potentially develop resistance to the immune response. Cancer Immunotherapy possesses only kinetic energy. This cycle Cycle The type of signal that ends the inspiratory phase delivered by the ventilator Invasive Mechanical Ventilation repeats as the pendulum swings back and forth. This phenomenon is similar to the changes in potential and kinetic energies that occur due to the effect of gravity on a free-falling object, or due to the compression Compression Blunt Chest Trauma/stretching oscillation of a spring.

Energy conservation of pendulums

Image by Lecturio.

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