Topicm7efa12bab204400b_1528449000663_0Topic

Simple vibrating motion and quantities describing it, examples of vibrating motion

Levelm7efa12bab204400b_1528449084556_0Level

Second

Core curriculumm7efa12bab204400b_1528449076687_0Core curriculum

VIII. Vibrating motionvibrating motionVibrating motion and waves. The student:

1) describes the periodic movement of the pendulum; uses the concepts of amplitude, period and frequency with their units to describe the periodic motion;

3) determines the amplitude and period of vibrations based on the graph of the distance versus time.

Timingm7efa12bab204400b_1528449068082_0Timing

45 minutes

General learning objectivesm7efa12bab204400b_1528449523725_0General learning objectives

Discussing the basic quantities describing the vibrating motion.

Key competencesm7efa12bab204400b_1528449552113_0Key competences

1. Gives examples of vibrating motion.

2. Explains the vibrating motion.

3. Defines the concepts of equilibrium position, deflection from the equilibrium position, amplitude, period and frequency of vibrations.

4. Uses known concepts in typical and new situations.

Operational (detailed) goalsm7efa12bab204400b_1528450430307_0Operational (detailed) goals

The student:

- is able to define the basic terms associated with a simple vibrating movement,

- applies the acquired concepts to solve the problems.

Methodsm7efa12bab204400b_1528449534267_0Methods

1. Discussion presenting new knowledge.

2. Conversational lecture.

Forms of workm7efa12bab204400b_1528449514617_0Forms of work

1. Work with the book.

2. Solving problem tasks using the learned concepts.

Lesson stages

Introductionm7efa12bab204400b_1528450127855_0Introduction

Observing the movement of the weight suspended on the spring, answer the following questions.

1. Is the weight movement a periodic or repetitive motion in time?
2. Is there a specific position of the weight around which this movement takes place?
3. Does the weight move with constant acceleration during vibrations?

Conclusion:

The weight, which is in a periodic motion, is moving around a characteristic point known as the position of the balance in a non‑uniformly variable motion.

Procedurem7efa12bab204400b_1528446435040_0Procedure

The weight suspended on the spring after precipitating it from the equilibrium position performs simple harmonic vibrations.
It is a periodic movement that repeats in time.
There is a characteristic point around which this movement takes place.
This characteristic point is called the position of balance.

Selected examples of vibrating motionvibrating motionvibrating motion.

[Illustration 1]

The weight suspended on the thread performs harmonic motions.

[Illustration 2]

The clock pendulum also performs a vibrating motion.

[Illustration 3]

The weight suspended on the spring after precipitation from the equilibrium position also performs harmonic vibrations.

A weight was hung on the spring, which was then set in a vibrating motion. During vibrations, the position of the weight was recorded and the results obtained were marked on the graph. Below are some slides from the experiment.

[Slideshow]

Definitions:

The deflection of the body from the equilibrium positionequilibrium positionequilibrium position is the distance of the body from the position of equilibrium. Usually it is marked with the letter x.

The amplitude of vibration is the maximum displacement of a vibrating body from its position of rest. Usually it is marked with the letter A.

The vibration periodvibration periodvibration period is the shortest time after which the movement begins to repeat. It is a time needed to perform one full vibration.

Comment:

When we want to determine practically the period of vibrations, the best method of it is to determine the time t needed to perform n full vibrations. The vibration periodvibration periodvibration period is then determined from the formula:

Vibration frequency is the number of vibrations performed within one second:

The unit of frequency is $[\frac{1}{s}=Hz]$.

Open the applet showing simple harmonic oscillationsharmonic oscillationsharmonic oscillations.

[Geogebra applet]

[Gallery]

Task

1. Set XIndeks dolny 01₀₁ = XIndeks dolny 02₀₂ = 0.
2. Determine TIndeks dolny 1₁ = TIndeks dolny 2₂.
3. Observe how the vibration plots change depending on the amplitudes AIndeks dolny 1₁ and AIndeks dolny 2₂.
4. Explain how the vibration amplitude influences the shape of the graphs.

Task

1. Set XIndeks dolny 01₀₁ = XIndeks dolny 02₀₂ = 0.
2. Determine AIndeks dolny 1₁ = 2AIndeks dolny 2₂.
3. Observe how the vibration plots change depending on TIndeks dolny 1₁ and TIndeks dolny 2₂.
4. Explain how the frequency of one and the other body can be determined on the basis of the graph.

Task

1. Set AIndeks dolny 1₁ = AIndeks dolny 2₂.
2. Determine TIndeks dolny 1₁ = TIndeks dolny 2₂.
3. Observe how the vibration plots change depending on XIndeks dolny 01₀₁ and XIndeks dolny 02₀₂.
4. Explain if the change of XIndeks dolny 01₀₁ or XIndeks dolny 02₀₂ affects the vibration frequency of both bodies.

Lesson summarym7efa12bab204400b_1528450119332_0Lesson summary

The vibration movement takes place around a point called the equilibrium position.

The quantities describing the vibration motion are:

The amplitude of vibrations A - the maximum displacement of a vibrating body from its position of rest; unit - meter [m].

The vibration period T - the time needed for one full vibration; the unit - second [s].

The vibration frequency f - number of vibrations per unit of time; unit - hertz [Hz].

The frequency and the period are related to each other in the following way: $f=\frac{1}{T}$.

The examples of vibrating systems are: the mathematical pendulum and the spring weight that perform harmonic oscillations.

The graph of the time dependence in the harmonic motion is a sinusoid. From this graph you can read the amplitude and period of vibration.

Selected words and expressions used in the lesson plan

harmonic oscillationsharmonic oscillationsharmonic oscillations

vibration periodvibration periodvibration period

amplitude of vibrationsamplitude of vibrationsamplitude of vibrations

frequency of vibrationsfrequency of vibrationsfrequency of vibrations

mathematical pendulummathematical pendulummathematical pendulum

pendulum on the springpendulum on the springpendulum on the spring

equilibrium positionequilibrium positionequilibrium position

elongationelongationelongation

harmonic motionharmonic motionharmonic motion

vibrating motionvibrating motionvibrating motion