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

울퉁불퉁한 도로를 달리는 자동차를 생각해 보십시오. 여기서 자동차의 서스펜션은 자동차의 질량을 바퀴에 연결하는 스프링과 같은 역할을 합니다.
움직이는 동안 스프링은 압축되고 팽창하여 간단한 조화 운동을 실행합니다.
자동차가 특정 진폭으로 튕긴다고 가정하면, 자동차의 최대 속도는 얼마이며, 초기 위치와 평형 위치 사이의 중간에 있다면 총 에너지는 어떻게 변할까요?
단순 조화 운동의 총 에너지는 보존된 상태로 유지됩니다. 항의 재배열은 모션 중 모든 위치에서 속도를 제공합니다.
최대 속도는 자동차의 스프링이 평형 위치를 통과할 때 발생합니다. 알려진 수량을 대체하면 최대 속도가 얻어집니다. 그런 다음 평형 위치에서의 총 에너지는 알려지고 계산된 양을 대체하여 얻습니다.
자동차의 스프링이 최대 진폭에서 절반 진폭으로 이동하면 속도는 음수로 간주됩니다. 총 에너지는 속도를 알려진 다른 양으로 대체하여 계산됩니다.
두 경우 모두 단순 조화 운동의 총 에너지는 동일합니다.

<ol>
	<li>Young, H.D. 및 Freedman, RA (2012). 현대 물리학을 가진 대학 물리학. 샌프란시스코, 캘리포니아 : 피어슨. 446쪽.</li>
	<li>오픈스택스. (2019). 대학 물리학 Vol. 1. [웹 버전]. 754쪽. 원본 주소 <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>

문제 해결: 단순 조화 운동의 에너지

단단조화운동(SHM)은 시간과 위치에 따른 주기적인 운동의 일종으로, 물체가 일정한 진폭과 주파수로 평형 위치를 중심으로 앞뒤로 진동합니다. SHM에서는 위치 에너지와 운동 에너지 사이에 지속적인 교환이 이루어지며, 이로 인해 물체가 진동하게 됩니다.

자동차 충격 흡수 장치의 스프링을 생각해 보세요. 바퀴에 부착된 스프링은 자동차가 울퉁불퉁한 도로를 주행하는 동안 단순 조화 운동을 실행합니다. 스프링에 가해지는 힘은 보존적이며 스프링이 늘어나거나 압축될 때 위치 에너지가 저장됩니다. 이 경우, 스프링에 부착된 바퀴는 스프링의 힘이 운동과 평행하게 작용하면서 1차원으로 진동합니다. 평형 위치에서 스프링에 저장된 위치 에너지는 0입니다. 소산력이 없을 경우 총에너지는 위치에너지와 운동에너지의 합으로 다음과 같이 표현된다.

<img alt="Equation1" src="/files/ftp_upload/14743/14743_Equation_1.svg" style="text-align: center; height: 35px;" />

단순 조화 운동의 총 에너지는 운동 중 모든 지점에서 시스템에 대해 보존되며 진폭의 제곱에 비례합니다.

단순 조화 운동의 총 에너지 방정식은 속도, 위치 및 총 기계적 에너지 사이의 유용한 관계를 나타냅니다. 이 방정식은 시간에 관계 없이 위치, 속도, 가속도 간의 관계가 필요한 문제에 사용할 수 있습니다. 에너지 보존 방정식에는 변위와 속도가 포함되므로 상황에서 변위와 속도의 부호를 유추해야 합니다. 예를 들어, 몸체가 평형 위치에서 가장 큰 양의 변위 지점을 향해 이동하는 경우 변위와 속도는 양의 값입니다.

단순 조화 운동의 에너지를 연구하는 것은 물리학과 공학 분야에서 진동 시스템의 동작을 이해하는 데 필수적입니다.

단순 조화 운동(SHM)은 물체가 평형 주위를 앞뒤로 진동하는 시간과 장소의 주기적 운동의 한 유형입니다.

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Education

교육

JoVE Core

Physics

물리학

Unit 1

제1단원

Oscillations

진동

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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...

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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.

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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...

/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...

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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

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

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3.2.9-12.O

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3.2.9-12.R

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3.2.9-12.S

Energy Conservation in Simple Harmonic Motion

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