Dao động điều hòa: Chu kì. Tần số. Tần số góc. Vận tốc và gia tốc của vật dao động điều hòa

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3 Dec 202004:35

Summary

TLDRIn this lesson, students explore the relationship between simple harmonic motion and uniform circular motion. The teacher uses a model of circular motion and its shadow to introduce key concepts such as amplitude, frequency, period, velocity, and acceleration. The lesson covers how the period is the time for a complete oscillation, the inverse relationship between period and frequency, and how angular velocity (omega) relates to harmonic motion. Students learn the formulas for velocity and acceleration in oscillatory motion, recognizing that acceleration always points toward equilibrium, even as velocity changes direction during the motion.

Takeaways

📚 The lesson explores the connection between harmonic oscillation and uniform circular motion.
🔄 A full oscillation is completed when the shadow of a uniformly moving object returns to its original position.
⏱️ The period of oscillation is defined as the time taken for one complete oscillation or one full rotation in circular motion.
🔢 The frequency is the inverse of the period, indicating the number of oscillations in a unit of time.
🔧 Frequency is measured in hertz (Hz), representing oscillations per second.
🌀 Angular velocity (omega) is the rate of change of angular displacement during uniform circular motion.
🧮 In harmonic motion, angular velocity is known as angular frequency and is involved in the oscillation equation.
🚀 Velocity changes with position: it's zero at the extremities and maximum at the equilibrium position.
⚖️ Acceleration is always directed toward the equilibrium position and is proportional to the displacement.
📊 The relationship between acceleration and displacement is represented by a = -ω²x, showing acceleration as a restoring force.

Q & A

What is the relationship between uniform circular motion and harmonic oscillation as introduced in the lesson?
-The relationship between uniform circular motion and harmonic oscillation is demonstrated through the model where the shadow of an object in uniform circular motion creates a harmonic oscillation. The key quantities involved include amplitude and initial phase.
What is the definition of the period (T) in harmonic oscillation?
-The period (T) is defined as the time taken for an object to complete one full oscillation and return to its initial position and direction. In terms of uniform circular motion, it is the time taken for the object to complete one full revolution.
How is the period of oscillation calculated if the object completes 'n' full oscillations in a given time?
-The period (T) is calculated by dividing the time (Δt) by the number of full oscillations (n): T = Δt / n.
What is the frequency (f) of oscillation, and how is it related to the period?
-The frequency (f) of oscillation represents the number of oscillations per unit of time and is calculated as the inverse of the period: f = 1/T. It is measured in Hertz (Hz), which is equivalent to oscillations per second.
What is angular velocity (ω) in uniform circular motion, and how is it connected to harmonic oscillation?
-Angular velocity (ω) is the rate of change of the angle swept by an object in uniform circular motion and is given by ω = 2π/T. In harmonic oscillation, ω is referred to as angular frequency and plays a key role in the oscillation equation X = A cos(ωt + φ).
How is the velocity of an object in harmonic oscillation related to its displacement?
-The velocity of the object in harmonic oscillation is the time derivative of displacement (x). At the extremes (amplitude), velocity is zero, and at the equilibrium position, velocity is at its maximum (v = ωA).
How does acceleration behave in harmonic oscillation?
-The acceleration of the object in harmonic oscillation is the second derivative of displacement and is proportional to displacement but in the opposite direction. The equation is a = -ω²x, meaning acceleration always points towards the equilibrium position.
How do velocity and acceleration vectors behave in harmonic oscillation?
-In harmonic oscillation, the velocity vector is always directed along the direction of motion. In contrast, the acceleration vector always points towards the equilibrium position, even when the object is moving away from it.
What happens to the velocity and acceleration when the object moves from the amplitude position towards the equilibrium?
-As the object moves from the amplitude (extreme) position towards the equilibrium, its velocity increases, and both velocity and acceleration point towards the equilibrium position.
What changes occur in the velocity and acceleration when the object moves from the equilibrium to the amplitude position?
-When the object moves from the equilibrium to the amplitude position, the velocity decreases, and while the velocity vector points towards the amplitude, the acceleration vector still points towards the equilibrium position.

Outlines

00:00

🔍 Introduction to Circular Motion and Harmonic Oscillation

This paragraph introduces the concept of harmonic oscillation and its connection to uniform circular motion. Using the model of a circular moving object and its shadow, students are familiarized with terms such as amplitude (A) and phase (phi). The lesson aims to further explore other important quantities like frequency, period, velocity, and acceleration.

🌀 Understanding the Period of Oscillation

This section explains how a full oscillation corresponds to one full rotation in uniform circular motion, bringing the object back to its original position. The time taken for one full oscillation is referred to as the period. The period can also be calculated if the object performs multiple oscillations in a given time interval (Delta T), according to a specific formula.

⏳ Calculating Frequency and Its Relationship with Period

Here, the concept of frequency is introduced as the inverse of the period. Frequency, which tells how many oscillations occur in a unit of time, is measured in hertz (Hz), defined as 'per second.' The paragraph emphasizes that frequency and period are inversely related quantities.

⚡ Angular Velocity in Circular Motion

This part explains how during one period of circular motion, the object sweeps through an angle, denoted as 'theta' (θ), which helps define the concept of angular velocity (omega). This angular velocity is also found in the equation for harmonic motion: X = A cos (omega t + phi), where omega is termed the angular frequency, a key parameter in both circular and harmonic motions.

🚀 Observing the Speed of Oscillating Motion

Students are encouraged to observe the speed of the shadow in the model. The shadow accelerates as it approaches the equilibrium position and decelerates after passing through it, with the motion reversing after reaching the extreme points (amplitude). This introduces the concept of varying velocity in harmonic motion.

📝 Deriving Velocity in Harmonic Motion

In this paragraph, the instantaneous velocity in harmonic motion is derived as the derivative of displacement. At the extremes, the velocity becomes zero, while at the equilibrium position, it reaches its maximum value, which is omega times the amplitude (A). These results align with observations made earlier.

⚖️ Exploring Acceleration in Harmonic Motion

By differentiating the velocity equation, the formula for acceleration is obtained. The paragraph highlights the direct relationship between acceleration and displacement, where acceleration is proportional to the negative displacement and is expressed as a = -omega^2 * X. This introduces the interplay between velocity, acceleration, and displacement in harmonic oscillation.

🎯 Understanding the Direction of Velocity and Acceleration

This section explains the direction of velocity and acceleration during oscillatory motion. When the object moves toward the equilibrium position, both velocity and acceleration point toward the center. Conversely, when moving away from the equilibrium, velocity and acceleration have opposite directions.

📚 Recap of Key Concepts in Harmonic Motion

The final paragraph summarizes the key quantities students have learned in the lesson, including amplitude, initial phase, period, frequency, angular frequency, velocity, and acceleration. Students are encouraged to thoroughly understand and memorize the definitions and equations of these fundamental properties in harmonic motion.

Mindmap

Keywords

💡Oscillatory Motion

Oscillatory motion refers to the repeated back-and-forth movement of an object between two points. In the video, this is exemplified through the periodic movement of the shadow of an object moving in uniform circular motion. The concept is crucial to understanding how certain parameters like amplitude, period, and frequency govern such motion.

💡Uniform Circular Motion

Uniform circular motion describes the motion of an object traveling in a circular path at constant speed. In the video, the model used shows how the shadow of a uniformly rotating object helps to visualize oscillatory motion, highlighting the connection between these two types of movements.

💡Amplitude (A)

Amplitude is the maximum displacement of an oscillating object from its equilibrium position. It represents the 'extent' of the oscillation. The video emphasizes amplitude as one of the key parameters of oscillatory motion, represented as 'A' in the equations of motion.

💡Period (T)

The period is the time it takes for an oscillating object to complete one full cycle of motion. In the context of the video, the period is related to the time required for the shadow of an object in uniform circular motion to return to its original position after one full oscillation.

💡Frequency (f)

Frequency refers to the number of oscillations that occur in a unit of time, typically measured in Hertz (Hz). The video defines frequency as the reciprocal of the period, explaining that it tells us how many complete oscillations occur per second.

💡Angular Velocity (ω)

Angular velocity, denoted as 'ω', measures the rate at which an object rotates around a circular path. In the video, it is introduced in the context of both circular and oscillatory motion. It also appears in the equation for oscillatory motion, where it relates to how quickly the angle changes in time.

💡Instantaneous Velocity

Instantaneous velocity refers to the speed and direction of an object at a specific moment in time. In the video, it is derived as the derivative of displacement with respect to time, showing how velocity changes as the object moves from the extreme positions toward equilibrium.

💡Acceleration

Acceleration describes how the velocity of an object changes over time. In the context of oscillatory motion, the video explains that acceleration is proportional to the displacement of the object but in the opposite direction, governed by the equation a = -ω²x, where ω is the angular velocity and x is the displacement.

💡Phase (Φ)

Phase refers to the specific point in the oscillation cycle that an object is at a given time. The video uses phase in the context of the equation for simple harmonic motion (x = A cos(ωt + Φ)), where it helps to determine the position of the oscillating object at any given moment.

💡Harmonic Oscillation

Harmonic oscillation is a type of oscillatory motion where the restoring force is directly proportional to displacement, resulting in a sinusoidal motion. In the video, the shadow of the object in uniform circular motion is shown to perform harmonic oscillation, demonstrating this principle.

Highlights

Introduction to the relationship between simple harmonic motion and circular motion through the model of a uniformly rotating object and its shadow.

Understanding amplitude and angular velocity through the shadow model.

Exploration of new quantities such as frequency, period, velocity, and acceleration.

Observation that the shadow on a horizontal plane completes a full oscillation when the object completes one circular motion.

Definition of the period of oscillation as the time taken for one complete oscillation.

Explanation of how the period is calculated using the formula for the number of complete oscillations.

Introduction to the concept of frequency as the number of oscillations per unit time.

Inverse relationship between frequency and period explained.

Discussion on how an object sweeps an angle during uniform circular motion.

Introduction to angular velocity as a characteristic quantity for the rate of change of the angle.

Angular velocity is also present in the oscillation equation X = A cos(Omega t + Phi).

Demonstration of how the shadow's speed changes as it moves from the boundary to the equilibrium position.

Explanation of how velocity is maximum at the equilibrium position and decreases as it moves away.

Derivation of the instantaneous velocity formula using the definition of velocity as the derivative of displacement.

Introduction to the concept of centripetal acceleration and its relationship with displacement.

Observation that velocity and acceleration have the same direction when moving towards the equilibrium position and opposite directions when moving away.

Vector nature of velocity and acceleration, with velocity following the direction of motion and acceleration always pointing towards the equilibrium position.

Summary of the basic quantities of simple harmonic motion learned, including amplitude, initial phase, period, angular velocity, velocity, and acceleration.

Emphasis on memorizing the definitions and formulas for calculating these quantities.

Conclusion and farewell, encouraging students to study the material carefully.