Name: energy conservation in simple harmonic motion
Uploaded: 2023-05-16T08:18:41.000Z
Duration: 1 min 17 s
Channel: JoVE

K12 Standards

Pennsylvania State Academic Standards

9-12

Physical Science

Energy

3.2.9-12.P

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

Problem Solving: Energy in Simple Harmonic Motion

ضع في اعتبارك سيارة تتحرك على طريق وعر. هنا ، يعمل تعليق السيارة مثل زنبرك يربط كتلة السيارة بالعجلات.
أثناء حركته ، يضغط الزنبرك ويتوسع ، مما يؤدي إلى تنفيذ حركة توافقية بسيطة.
لنفترض أن السيارة ترتد بسعة معينة؛ ما هي سرعتها القصوى، وكيف تتغير طاقتها الكلية إذا كانت في منتصف المسافة بين موقعها الأولي والتوازن؟
تظل الطاقة الكلية في الحركة التوافقية البسيطة محفوظة. تعطي إعادة ترتيب المصطلحات السرعة في أي موضع أثناء الحركة.
تحدث السرعة القصوى عندما يمر زنبرك السيارة عبر موضع التوازن. عند استبدال الكميات المعروفة ، يتم الحصول على السرعة القصوى. ثم يتم الحصول على الطاقة الإجمالية في وضع التوازن عن طريق استبدال الكميات المعروفة والمحسوبة.
عندما يتحرك زنبرك السيارة من الحد الأقصى إلى نصف السعة ، تعتبر السرعة سالبة. يتم حساب الطاقة الإجمالية عن طريق استبدال السرعة بكميات أخرى معروفة.
في كلتا الحالتين ، تكون الطاقة الكلية للحركة التوافقية البسيطة هي نفسها.

<ol>
	<li>يونغ ، إتش دي وفريدمان ، ر. (2012). الفيزياء الجامعية مع الفيزياء الحديثة. سان فرانسيسكو ، كاليفورنيا: بيرسون. ص 446.</li>
	<li>OpenStax. (2019). جامعة الفيزياء المجلد 1. [نسخة الويب]. ص 754. تم الاسترجاع من <a href="https://openstax.org/books/university-physics-volume-1/pages/15-2-energy-in-simple-harmonic-motion">https://openstax.org/books/university-physics-volume-1/pages/15-2-energy-in-simple-harmonic-motion</a></li>
</ol>

حل المشكلات: الطاقة في الحركة التوافقية البسيطة

<div style="direction:rtl">الحركة التوافقية البسيطة (SHM) هي نوع من الحركة الدورية في الزمن والموضع، حيث يتأرجح الجسم ذهابًا وإيابًا حول موضع توازن بسعة وتردد ثابتين. في SHM، هناك تبادل مستمر بين الطاقة الكامنة والطاقة الحركية، مما يؤدي إلى تذبذب الجسم. فكر في الزنبرك الموجود في ممتص الصدمات في السيارة. ينفذ الزنبرك المتصل بالعجلة حركة توافقية بسيطة أثناء تحرك السيارة على طريق وعر. القوة المؤثرة على الزنبرك متحفظة، ويتم تخزين الطاقة الكامنة عند تمديد الزنبرك أو ضغطه. في هذه الحالة، تتأرجح العجلة المرتبطة بالزنبرك في بُعد واحد، حيث تعمل قوة النابض بالتوازي مع الحركة. في وضع التوازن، تكون طاقة الوضع المخزنة في الزنبرك صفرًا. إذا لم تكن هناك قوى تبديد، فإن الطاقة الإجمالية هي مجموع الطاقة الكامنة والطاقة الحركية ويتم التعبير عنها على النحو التالي: <img alt="Equation1" src="/files/ftp_upload/14743/14743_Equation_1.svg" style="text-align: center; height: 35px;"> تظل الطاقة الكلية في الحركة التوافقية البسيطة محفوظة للنظام عند كل نقطة أثناء الحركة وتتناسب مع مربع السعة. تمثل معادلة الطاقة الإجمالية في الحركة التوافقية البسيطة علاقة مفيدة بين السرعة المتجهة المتجهة المتجهة المتجهة والموضع والطاقة الميكانيكية الإجمالية. يمكن استخدام هذه المعادلة إذا كانت المسألة تتطلب وجود علاقة بين الموقع والسرعة المتجهة المتجهة المتجهة المتجهة والتسارع دون الرجوع إلى الزمن. بما أن معادلة حفظ الطاقة تتضمن الإزاحة والسرعة المتجهة المتجهة، فيجب استنتاج علامات الإزاحة والسرعة المتجهة المتجهة المتجهة المتجهة من الموقف. على سبيل المثال، إذا تحرك الجسم من موضع التوازن نحو نقطة الإزاحة الموجبة الأكبر، تكون الإزاحة والسرعة المتجهة المتجهة المتجهة المتجهة موجبة. إن دراسة الطاقة في الحركة التوافقية البسيطة أمر حيوي لفهم سلوك الأنظمة المتذبذبة في الفيزياء والهندسة.</div>

الحركة التوافقية البسيطة (SHM) هي نوع من الحركة الدورية في الوقت والموضع ، حيث يتأرجح الجسم ذهابا وإيابا حول التوازن ...

/files/media/science-education/science-education-thumbs/14743.jpg

Education

JoVE Core

Physics

الفيزياء

Unit 1

الوحدة الأولى

Oscillations

تذبذبات

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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 one dimension, and the spring force acts parallel to the motion. The force is conservative, so energy is stored as potential energy when the spring is stretched or compressed.

At the equilibrium position, the spring’s potential energy is zero. If no dissipative forces are present, the total mechanical energy stays constant throughout the motion. This total energy is the sum of kinetic energy and potential energy. In SHM, it is also proportional to the square of the amplitude.

The total energy equation in SHM links position, velocity, and mechanical energy. It is useful when a problem asks for a connection between position, velocity, and acceleration without using time. Because the equation includes displacement and velocity, the signs must be inferred from the motion. For example, if the object moves from equilibrium toward the greatest positive displacement, both displacement and velocity are positive.

Understanding energy in simple harmonic motion helps explain how oscillating systems behave in physics and engineering.

Energy Conservation in Simple Harmonic Motion

energy-conservation-in-simple-harmonic-motion

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...

/files/media/science-education/science-education-thumbs/12762.jpg

characteristics of simple harmonic motion

characteristics-of-simple-harmonic-motion

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...

/files/media/science-education/science-education-thumbs/12763.jpg

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.

/files/media/science-education/science-education-thumbs/12765.jpg

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...

/files/media/science-education/science-education-thumbs/14718.jpg

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

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...

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...

/files/media/science-education/science-education-thumbs/12768.jpg

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...

/files/media/science-education/science-education-thumbs/14719.jpg

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,...

/files/media/science-education/science-education-thumbs/14720.jpg

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...

/files/media/science-education/science-education-thumbs/12772.jpg

key terms and concepts

Learn how kinetic and potential energy change in simple harmonic motion, with a spring example and the energy equation used to solve SHM problems.

Energy Conservation in Simple Harmonic Motion | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

3.2.9-12.Q

Louisiana Student Standards for Science

High School

HS-PS3-2

HS-PS3-3

Next Generation Science Standards

HS-PS3-1

Florida Department of Education

SC.912.P.10

SC.912.P.10.6

Texas Essential Knowledge and Skills

PHYS.8

PHYS.8.A

The universe is composed of matter in different forms, and all forms of matter contain energy.&#160; The different forms of energy on Earth originate from ...

Energy Basics

Chemical reactions, such as those that occur when you light a match, involve changes in energy as well as matter.

Chemical changes and their accompanying ...

Conservation of Mechanical Energy

The mechanical energy E of a system is the sum of its potential energy U and the kinetic energy K of the objects within it. What happens to this ...

/files/media/science-education/science-education-thumbs/14702.jpg

Conservation of Energy

The terms &#39;conserved quantity&#39;&#160;and &#39;conservation law&#39;&#160;have specific scientific meanings in physics, which differ from the ...

Conservation of Energy: Application

When solving problems using the energy conservation law, the object (system) to be studied should first be identified. Often, in applications of energy ...

Students who demonstrate understanding can create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known. Clarifying Statement: Emphasis is on explaining the meaning of mathematical expressions used in the model. Assessment Boundry: Assessment is limited to basic algebraic expressions or computations; to systems of two or three components; and to thermal energy. kinetic energy, and/or the energies in gravitational, magnetic, or electric fields.

3.2.9-12.O

Types of Potential Energy

Potential energy is also known as energy at rest or stored energy. Common types of potential energy include the gravitational potential energy stored in ...

Types of Kinetic Energy

The amount of kinetic energy of an object depends on its mass and speed. Consider two balls of different masses rolling down an inclined plane at the same ...

Kinetic Energy - I

It&#8217;s plausible to suppose that the greater the velocity of a body, the greater effect it could have on other bodies. This does not depend on the ...

Kinetic Energy - II

The kinetic energy of a particle is one-half of the product of the particle&#8217;s mass and the square of its speed. Note that just as Newton&#8217;s ...

Gravitational Potential Energy

Potential energy is not just a property of each object, but also a property of the interactions between objects in a chosen system. For each type of ...

Elastic Potential Energy

Elastic potential energy is the energy stored as a result of the deformation of an elastic object, such as the stretching of a spring. An object is ...

Energy in Simple Harmonic Motion

To determine the energy of a simple harmonic oscillator, consider all the forms of energy it can have during its simple harmonic motion. According to ...

Simple harmonic motion (SHM) is a type of periodic motion in time and position, in which an object oscillates back and forth around an equilibrium ...

Kinetic and Potential Energy of a Wave

All forms of waves carry energy; this is directly visualized in nature. For instance, the waves of earthquakes are so intense that they can shake huge ...

Energy and Power of a Wave

The total energy associated with a wavelength is the sum of the potential energy and the kinetic energy. The average rate of energy transfer associated ...

Energy Carried By Electromagnetic Waves

Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes, this energy is obvious, such as in the summer sun&#39;s ...

Students who demonstrate understanding can develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative positions of particles (objects). Clarifying Statement: Examples of phenomena at the macroscopic scale could include the conversion of kinetic energy to thermal energy, the energy stored due to position of an object above the earth, and the energy stored between two electrically-charged plates. Examples of models could include diagrams, drawings, descriptions, and computer simulations. Assessment Boundry: N/A

What is Energy?

The universe is composed of matter in different forms, and all forms of matter contain energy. &#xA0;The different forms of energy on Earth originate from ...

Energy Transfer in Chemical Reactions

Chemical reactions require sufficient energy to cause the matter to collide with enough precision and force that old chemical bonds can be broken and new ...

Students who demonstrate understanding can design, build and refine a device that works within given constraints to convert one form of energy into another form of energy. Clarifying Statement: Emphasis is on both qualitative and quantitative evaluations of devices. Examples of devices could include Rube Goldberg devices, wind turbines, solar cells, solar ovens, and generators. Examples of constraints could include use of renewable energy forms and efficiency. Assessment Boundry: Assessment for quantitative evaluations is limited to total output for a given input. Assessment is limited to devices constructed with materials provided to students.

Students who demonstrate understanding can plan and conduct an investigation to provide evidence that the transfer of thermal energy when two components of different temperature are combined within a closed system results in a more uniform energy distribution among the components in the system (second law of thermodynamics). Clarifying Statement: Emphasis is on analyzing data from student investigations and using mathematical thinking to describe the energy changes both quantitatively and conceptually. Examples of investigations could include mixing liquids at different initial temperatures or adding objects at different temperatures to water. Assessment Boundry: Assessment is limited to investigations based on materials and tools provided to students.

3.2.9-12.R

force-and-potential-energy-in-one-dimension

Force and Potential Energy in One Dimension

Force can be calculated from the expression for potential energy, which is a function of position. The component of a conservative force, in a particular ...

force-and-potential-energy-in-three-dimensions

Force and Potential Energy in Three Dimensions

Consider a particle moving under the action of a conservative force that has components along each coordinate axis. Each component of force is a function ...

Energy Diagrams - I

The dynamics of a mechanical system can be easily understood by interpreting a potential energy diagram. Since energy is a scalar quantity, the ...

Energy Diagrams - II

Energy diagrams are important to understand the dynamics of a system. The topology of an energy diagram helps illustrate the equilibrium points of the ...

Students who demonstrate understanding can develop and use a model of two objects interacting through electric or magnetic fields to illustrate the forces between objects and the changes in energy of the objects due to the interaction. Clarifying Statement: Examples of models could include drawings, diagrams, and texts, such as drawings of what happens when two charges of opposite polarity are near each other. Assessment Boundry: Assessment is limited to systems containing two objects.

3.2.9-12.S

Energy Conservation in Simple Harmonic Motion

Description

Transcript

More videos by Standards