Name: how mass and spring stiffness set oscillation
Uploaded: 2025-05-22T09:06:09.000Z
Duration: 1 min 17 s
Channel: JoVE

K12 Standards

ohios-learning-standards-and-model-curriculum

Ohio’s Learning Standards and Model Curriculum

9-12

Physical Science

Forces And Motion

Forces

Frequency of Spring-Mass System

水平スプリングに結合された質量mのブロックを、摩擦のない表面上に配置するとします。
ブロックにかかる正味の力は、ブロックの重量による力、垂直抗力、およびスプリングによる力の合計です。
重量と垂直抗力は大きさが等しく、方向が逆であるため、互いに打ち消し合い、正味の力はバネによる力と等しくなります。
ここで、力の大きさは変位の最初の力に比例します。このため、ばね質量システムは線形単純調和振動子と呼ばれます。
ニュートンの第二法則を使用すると、力は加速度で表すことができます。
加速度と変位の式を代入すると、角周波数の方程式が得られます。
角周波数は、発振期間にわたって2πとしても定義されます。
また、周期の逆数は振動の周波数です。
硬いスプリングは、急速な振動と短い周期を生み出します。それに比べて、重い物体は振動が鈍く、周期が長くなる傾向があります。

<ol>
	<li>OpenStaxです。(2019). 大学物理学 Vol.1.【Web版】。<a href="https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion"から取得>https://openstax.org/books/university-physics-volume-1/pages/15-1-simple-harmonic-motion</a></li>
</ol>

ばね-質量系の周波数

ばねに取り付けられた物体の単調和運動 (SHM) の興味深い特性の 1 つは、角周波数、動きの周期と周波数がばねの質量と力定数のみに依存することです。 動きの振幅や初期条件などの他の要因には依存しません。 運動方程式とニュートンの第 2 法則を使用して、角周波数、周波数、周期を求めることができます。

摩擦のない表面上のばね上のブロックを考えてみましょう。 質量には 3 つの力、すなわち重量、垂直力、およびバネによる力がかかります。 表面に垂直に作用する力は重量と垂直力の 2 つだけで、大きさが等しく反対方向です。 その結果、それらの合計はゼロになります。 表面に平行に作用する力はバネによる力だけであるため、正味の力はバネの力と等しくなければなりません。

フックの法則によれば、力と変形が十分に小さい限り、ばねの力の大きさは変位の 1 乗に比例します。 このため、バネ質量系は線形単調和振動子と呼ばれます。

ニュートンの第 2 法則の加速度と変位の式を代入すると、角周波数の方程式が得られます。

<img alt="Equation1" src="/files/ftp_upload/14718/14718_Equation_1.svg" style="text-align: center; height: 48px;" />

角周波数は力定数と質量のみに依存し、振幅には依存しません。 また、指定された関係を使用して振動周期とも関連付けられます。

<img alt="Equation2" src="/files/ftp_upload/14718/14718_Equation_2.svg" style="text-align: center; height: 40px;" />

周期も質量と力定数のみに依存します。 質量が大きいほど周期は長くなります。 ばねが硬いほど、周期は短くなります。 頻度は

です。

<img alt="Equation3" src="/files/ftp_upload/14718/14718_Equation_3.svg" style="text-align: center; height: 48px;" />

バネに取り付けられた物体の単純調和運動(SHM)の興味深い特性の1つは、角周波数、周期、および...

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

Education

JoVE Core

Physics

物理学

Unit 1

ユニット1

Oscillations

振動

KEY TERMS AND CONCEPTS

主要用語と概念

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 force.

The weight and normal force act perpendicular to the surface. They have equal magnitudes and opposite directions, so their sum is zero. The only force parallel to the surface is the spring force, so the net force is the force from the spring.

Hooke’s law describes that spring force when the deformation is small. It says the force is proportional to the first power of displacement. Because of this linear relationship, the spring-mass system is called a linear simple harmonic oscillator.

By substituting the expressions for acceleration and displacement into Newton’s second law, the angular frequency can be found. This angular frequency depends only on the mass and the force constant, not on amplitude. The period is related to angular frequency, and it also depends only on the mass and the force constant.

A larger mass gives a longer period. A stiffer spring gives a shorter period. The frequency follows from the period and describes how many oscillations occur in a given time.

How Mass and Spring Stiffness Set Oscillation

how-mass-and-spring-stiffness-set-oscillation

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...

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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

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...

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

Learn how a spring-mass system’s frequency, period, and angular frequency depend on mass and spring stiffness, not amplitude or initial conditions.

How Mass and Spring Stiffness Set Oscillation | JoVE K12

physics

JoVE Core 物理学は、運動学、力学、流体力学、静力学、波動、音響などの基礎的な物理学の理論と応用を、簡潔でわかりやすいアニメーション ビデオで紹介します。さらに、科学者の実際のビデオでは、世界中の今日の研究所で行われている古典的および独自の研究実験における関連概念の適用を示しています。

Forces, Momentum And Motion

newton-s-laws-applied-to-complex-problems

Newton's laws applied to complex problems

Elastic forces

Energy

Energy in springs

Tennessee Academic Standards

High School

motion-and-stability-forces-and-interactions

Motion and Stability: Forces and Interactions

PHYS.PS2.5

waves-and-their-applications-in-technologies-for-information-transfer

Waves and Their Applications in Technologies for Information Transfer

PHYS.PS4.2

Texas Essential Knowledge and Skills

PHYS.8

PHYS.8.A

Chemistry and Physics

IPC.6

IPC.6.C

Inertial Frames of Reference

Newton&#8217;s first law is usually considered to be a statement about reference frames. It provides a method for identifying a special type of reference ...

Non-inertial Frames of Reference

A reference frame accelerating or decelerating relative to an inertial frame is a non-inertial frame. To help understand this, consider what taking off in ...

Introduction to Scalars

Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, &#34;a class period lasts ...

Introduction to Vectors

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Position and Displacement

The position of an object defines its location relative to a convenient frame of reference at any particular time. A frame of reference is an arbitrary ...

Average Velocity

To calculate the other physical quantities in kinematics, we must introduce the time variable. The time variable allows us not only to state the position ...

Instantaneous Velocity - I

The average velocity during a time interval cannot tell us how fast or in what direction a particle is moving at any given time during the interval. To ...

Instantaneous Velocity - II

Instantaneous velocity is the quantity that measures how fast an object is moving along its path. In other words, the instantaneous velocity of an object ...

Average Acceleration

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Instantaneous Acceleration

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Position and Displacement Vectors

To describe the motion of an object, one should first be able to describe its position (where it is at any particular time). More precisely, the position ...

average-and-instantaneous-velocity-vectors

Average and Instantaneous Velocity Vectors

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Acceleration Vectors

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, &#916;v, therefore the ...

Direction of Acceleration Vectors

Acceleration occurs when velocity changes in magnitude (an increase or decrease in speed), direction, or both. Although acceleration is in the direction ...

velocity-and-position-by-graphical-method

Velocity and Position by Graphical Method

Velocity and position can be calculated from the known function of acceleration as a function of time. The total area under the acceleration-time graph ...

• Introduction to one-dimensional vectors 
• Displacement, velocity (constant, average and instantaneous) and acceleration 
• Interpreting position vs. time and velocity vs. time graphs

Content elaboration: The motion of an object depends on the observer's frame of reference and is described in terms of distance, position, displacement, speed, velocity, acceleration and time. Position, displacement, velocity and acceleration are all vector properties (magnitude and direction). All motion is relative to whatever frame of reference is chosen for there is no motionless frame from which to judge all motion. The relative nature of motion will be addressed conceptually, not mathematically. Non-inertial reference frames are excluded. Motion diagrams can be drawn and interpreted to represent the position and velocity of an object. Showing acceleration on motion diagrams is reserved for Physics. The displacement or change in position of an object is a vector quantity that can be calculated by subtracting the initial position from the final position $(\Delta x=x_{f}-x_{i})$. Displacement can be positive or negative depending upon the direction of motion. Displacement is not always equal to the distance travelled. Examples should be given where the distance is not the same as the displacement. Velocity is a vector quantity that represents the rate at which position changes. Average velocity can be calculated by dividing displacement (change in position) by the elapsed time $(v_{avg}=(x_{f}-x_{i})/(t_{f}-t_{i}))$. Velocity may be positive or negative depending upon the direction of motion. Velocity should be distinguished from speed, which is always positive. Provide examples of when the average speed is not the same as the average velocity. Objects that move with constant velocity have the same displacement for each successive time interval. While speeding up or slowing down and/or changing direction, the velocity of an object changes continuously, from instant to instant. The speed of an object at any instant (clock reading) is called instantaneous speed. Acceleration is a vector quantity that represents the rate at which velocity changes. Average acceleration can be calculated by dividing the change in velocity by elapsed time $(a_{avg}=(v_{f}-v_{i})/(t_{f}-t_{i}))$. At this grade level, it should be noted that acceleration can be positive or negative, but specifics about what kind of motions produce positive or negative accelerations will be addressed in Physics. Deceleration is an ambiguous term that should only be used when an object is slowing down. Care should be given to ensure students do not associate negative acceleration with only deceleration. [cite: 352, 357] Objects with negative acceleration could be increasing their speed. Objects that have no acceleration can either be standing still or be moving with constant velocity (speed and direction). Constant acceleration occurs when the change in an object's instantaneous velocity is the same for equal successive time intervals. Motion can be represented by position vs. time and velocity vs. time graphs. Specifics about the speed, direction and change in motion can be determined by interpreting such graphs. For this course, graphs will be limited to positive x-values and show only uniform motion involving segments of constant velocity or constant acceleration. Motion can be investigated by collecting and analyzing data in the laboratory and should include constant velocity as well as constant acceleration. Technology can enhance motion exploration and investigation through video analysis, the use of motion detectors and graphing data for analysis. Objects that move with constant velocity and have no acceleration form a straight line (not necessarily horizontal) on a position vs. time graph. Objects that are at rest will form a horizontal line on a position vs. time graph. Objects that are accelerating will show a curved line on a position vs. time graph. Velocity can be calculated by determining the slope of a position vs. time graph. Positive slopes on position vs. time graphs indicate motion in a positive direction. time graphs indicate motion in a negative direction. While it is important that students can construct graphs by hand, computer graphing programs or graphing calculators can also be used so more time can be spent on graph interpretation and analysis. Constant acceleration is represented by a straight line (not necessarily horizontal) on a velocity vs. time graph. Objects that have no acceleration (at rest or moving at a constant velocity) will have a horizontal line for a velocity vs. time graph. Average acceleration can be determined from the slope of a velocity vs. time graph. The details about motion graphs should not be taught as rules to memorize, but rather as generalizations that can be developed from interpreting the graphs.

Motion

Force

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Types of Forces

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Drawing Free-body Diagrams: Rules

The first step in describing and analyzing most phenomena in physics involves the careful drawing of a free-body diagram. Free-body diagrams are useful in ...

Free Body Diagrams: Examples

Solving problems that involve forces is easy using free-body diagrams. A free-body diagram is a sketch showing all the external forces that are acting on ...

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

One of the simpler characteristics of sliding friction is that it is parallel to the contact surfaces between systems, and is always in a direction that ...

Drag Force and Terminal Speed

An interesting force in everyday life is the force of drag on an object when it is moving in a fluid. Like friction, the drag force always opposes the ...

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Tension

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Mass and Weight

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Coulomb's Law

Experiments with electric charges have shown that if two objects each have an electric charge, they exert an electric force on each other. The magnitude ...

Magnetic Force

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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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Electric Field

Consider two point charges, each exerting Coulomb force on the other. It is possible to describe the Coulomb interaction via an intermediate step by ...

Magnetic Fields

A moving charge or a current creates a magnetic field in the surrounding space, in addition to its electric field. The magnetic field exerts a force on ...

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 ...

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 ...

• Force diagrams 
• Types of forces (gravity, friction, normal, tension) 
• Field model for forces at a distance

Content elaboration: Force is a vector quantity, having both magnitude and direction. Force diagrams are useful tools for visualizing and analyzing the forces acting on objects. The (SI) unit of force is a newton. One newton of net force will cause a 1 kg object to experience an acceleration of $1~m/s^{2}$. A newton can also be represented as $kg\cdot m/s^{2}$. The opportunity to measure force in the lab is provided (e.g., with a spring scale or a force probe). Normal forces and tension forces are introduced conceptually at this level. These forces and other forces introduced in prior grades (friction, drag, gravitational, electric and magnetic) can be used as examples of forces that affect motion. In this course, only forces in one dimension (positive and negative) will be addressed. The net force can be determined by one-dimensional vector addition. Gravitational force (weight) can be calculated from mass, but all other forces will only be quantified from force diagrams. Friction is a force that opposes motion. Kinetic friction (e.g., sliding, rolling), drag and static friction can be addressed conceptually. More quantitative study of friction forces, universal gravitational forces, elastic forces and electrical forces is reserved for Physics. A normal force exists between two solid objects when their surfaces are pressed together due to other forces acting on one or both objects (e.g., a solid sitting on or sliding across a table, a ladder leaning against a wall, a ball hitting a bat). A normal force is always a push directed at right angles from the surfaces of the interacting objects. A tension force occurs when a non-slack rope, wire, cord or similar device pulls on another object. In middle school, the concept of a field as a region of space that surrounds objects with the appropriate property (mass for gravitational fields, charge for electric fields, a magnetic object for magnetic fields) was introduced to explain gravitational, magnetic and electrical forces that occur over a distance. In high school, the field concept is further developed. The stronger the field, the greater the force exerted on objects placed in the field. The field of an object is always there even if the object is not interacting with anything else. The gravitational force (weight) of an object is proportional to its mass. Weight, $F_{g},$ can be calculated from the equation $F_{g}=mg,$ where g is the gravitational field strength of an object which is equal to 9.8 N/kg or $9.8~m/s^{2}$ on the surface of Earth.

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 ...

Newton's Third Law: Introduction

Whenever one body exerts a force on a second body, the first body experiences a force equal in magnitude and opposite in direction, to the force that it ...

Newton's Third Law: Examples

Newton&#39;s third law states that every action has an equal and opposite reaction. Consider a swimmer pushing off the side of a pool. They push against ...

• Objects at rest 
• Objects moving with constant velocity 
• Accelerating objects

Content elaboration: The focus of the content is to develop a conceptual understanding of the laws of motion to explain and predict changes in motion, not to name or recite a memorized definition. When the vector sum of the forces (net force, Fnet) acting on an object is zero, the object does not accelerate. For an object that is moving, this means the object will remain moving without changing its speed or direction. For an object that is not moving, the object will continue to remain stationary. An object will accelerate (increase or decrease its speed or change its direction of motion) when an unbalanced net force acts on it. The rate at which an object changes its speed or direction (acceleration) is proportional to the vector sum of the forces (net force, $F_{net}$) and inversely proportional to the mass $(a=F_{net}/m)$. These laws will be applied to systems consisting of a single object upon which multiple forces act. Vector addition will be limited to one dimension (positive and negative). While both horizontal and vertical forces can be acting on an object simultaneously, for this level, one of the dimensions must have a net force of zero. A force is an interaction between two objects. Both objects in the interaction experience an equal amount of force, but in opposite directions. Interacting force pairs are often confused with balanced forces. Interacting force pairs can never cancel each other out because they always act on different objects. Naming the force (e.g., gravity, friction) does not identify the two objects involved in the interacting force pair. Objects involved in an interacting force pair can be easily identified by using the format "A acts on B so B acts on A." For example, the truck hits the sign therefore the sign hits the truck with an equal force in the opposite direction. Earth pulls the book down so the book pulls Earth up with an equal force. In Physics, all laws will be applied to systems of many objects.

Dynamics (how forces affect motion)

How Mass and Spring Stiffness Set Oscillation

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