Lesson plan

Topicmff10d7c099ed20f3_1528449000663_0Topic

Determination of the period and frequency of vibration of the mathematical pendulum and the weight on the spring

Levelmff10d7c099ed20f3_1528449084556_0Level

Second

Core curriculummff10d7c099ed20f3_1528449076687_0Core curriculum

VIII. Vibrating motion and waves. The student:

9) experimentally:

a) determines the period and frequency of the periodic movement.

Timingmff10d7c099ed20f3_1528449068082_0Timing

45 minutes

General learning objectivesmff10d7c099ed20f3_1528449523725_0General learning objectives

Experimental determination of the frequency of vibrationsfrequency of vibrationsfrequency of vibrations, the period of the mathematical pendulummathematical pendulummathematical pendulum and the weight on the springspringspring.

Key competencesmff10d7c099ed20f3_1528449552113_0Key competences

1. Improving the skills of designing the experimental stand.

2. Determining the frequency and vibration periodvibration periodvibration period of the mathematical pendulum.

3. Determining the frequency and period of vibrations of the body suspended on a spring.

Operational (detailed) goalsmff10d7c099ed20f3_1528450430307_0Operational (detailed) goals

The student:

- designs an experimental stand to determine the period of vibration of the mathematical pendulum and the weightweightweight on the springspringspring,

- performs simple physical measurements and analyses the obtained results.

Methodsmff10d7c099ed20f3_1528449534267_0Methods

1. Applying theory in practice.

2. Applying the acquired knowledge in practice.

Forms of workmff10d7c099ed20f3_1528449514617_0Forms of work

1. Group work when performing video physical measurements.

2. Preparation of the obtained results and analysis of measurement errors.

Lesson stages

Introductionmff10d7c099ed20f3_1528450127855_0Introduction

Answer the following questions:

1. What is the mathematical pendulummathematical pendulummathematical pendulum?

2. What and how does the period of vibration of the mathematical pendulum depend on?

3. What is the springspringspring pendulum?

4. What and how does the period of vibrations of the weightweightweight on the spring depend on?

Proceduremff10d7c099ed20f3_1528446435040_0Procedure

Experiment 1

Analysis of the dependence of the vibration periodvibration periodvibration period of a mathematical pendulum on the length l of the threadthreadthread.

Research hypothesis:

The period of vibration of the mathematical pendulum is proportional to $\sqrt{l}$ .

Instruction:

1. Build a mathematical pendulum. You can use the following things.

Slideshow - what is needed to complete the experiment.

[Slideshow]

2. Determine the specified length l of the mathematical pendulum threadthreadthread.

3. Measure the time needed to complete n > 1 (for example n = 5) full vibration of the mathematical pendulum. It will be easier to read the right time if you use the video recording on your mobile phone.

4. Repeat steps 2 and 3 several times to eliminate the gross errorgross errorgross error.

5. Repeat the measurements for 15 different lengths of the mathematical pendulum between 20 cm and 200 cm.

Note:

While making the measurements, make sure that the amplitude of the mathematical pendulum oscillation is not too high.mff10d7c099ed20f3_1527752263647_0While making the measurements, make sure that the amplitude of the mathematical pendulum oscillation is not too high.

A suggested measurement table:

[Table 1]

Processing of the results:

1. For each measurement, determine the period of vibration.

2. Make a graph of the dependence between the vibration periodvibration periodvibration period T and the length l of the mathematical pendulummathematical pendulummathematical pendulum thread.

3. Based on the obtained graph, answer the question: Is the research hypothesis confirmed by the graph?

4. Present the possible sources of measurement uncertainty which occur during the experiment.

5. Compare the experimentally determined period with the theoretical formula:

T = 2 \cdot π \cdot \sqrt{\frac{l}{g}}

where:
T - vibration period [s],
l - thread length [m],
g - gravitational acceleration 10 $\frac{m}{s^{2}}$ .

Experiment 2

Investigation of the dependence between the vibrations period of the weightweightweight suspended on the springspringspring and the mass of the weight.

Research hypothesis:

The vibration periodvibration periodvibration period of the weight on the spring is proportional to $\sqrt{m}$ .

1. Build a spring pendulum. To do this, hang a spring on the support standsupport standsupport stand and then attach various weights.

[Illustration 1]

2. Attach a weight of mass m on the spring.

3. Measure the time needed to complete n the full vibrations of the weightweightweight on the springspringspring.

4. Repeat steps 2 and 3 several times to eliminate the gross errorgross errorgross error.

5. Repeat the measurements for 10 different weight masses on the spring from 20 g to 500 g.

A suggested measurement table:

[Table 2]

Processing of the results:

1. For each measurement, determine the period of vibration.

2. Make a graph of the dependence between the vibration periodvibration periodvibration period T and the length l of the mathematical pendulummathematical pendulummathematical pendulum thread.

3. Based on the obtained graph, answer the question: Is the research hypothesis confirmed by the graph?

4. Present the possible sources of measurement uncertainty which occur during the experiment.

5. Compare the experimentally determined period with the theoretical formula:

T = 2 \cdot π \cdot \sqrt{\frac{m}{k}}

where:
T - vibration period [s],
m - body weight [kg],
k - spring elasticity coefficient $[\frac{N}{kg}]$ .

You can determine the coefficient k using the dependence of the spring extension value on the mass of the weight suspended at the end of the spring:

k = | \frac{F}{x} | = | \frac{m \cdot g}{x} |

where:
F - force of gravity [N],
k - spring elasticity coefficient $[\frac{N}{kg}]$ ,
m - body weight [kg],
g - gravitational acceleration 10 $\frac{m}{s^{2}}$ ,
x - displacement from the equilibrium position [m].

Lesson summarymff10d7c099ed20f3_1528450119332_0Lesson summary

Mathematical pendulummathematical pendulumMathematical pendulum:

The conclusions from the measurements are following:

- the period of vibration of the mathematical pendulum depends on its length;
- longer duration of vibration corresponds to a longer length of the mathematical pendulum;
- when the length of the pendulum thread of the mathematical pendulum increases by n times, the period of vibration will increase $\sqrt{n}$ , e.g. when the thread length is increased by four times, the vibration period will increase only two times.

The weightweightweight on the springspringspring:

The conclusions from the measurements are following:

- the period of vibrations of the weight on the spring depends on its mass;
- greater mass corresponds to a higher value of the vibration period;
- when the mass of weights increases four times, the period of vibration will increase twice (this effect is possible if the mass of the spring is much smaller than the mass of the weight suspended on it).

Both experiences clearly confirm the validity of theoretical formulas for the period of vibration of the mathematical pendulum and the weight on the spring.

The dependence of the period of vibration of the mathematical pendulum on its length is expressed by the formula:

T = 2 \cdot π \cdot \sqrt{\frac{l}{g}}

where:
T - vibration period [s],
l - thread length [m],
g - gravitational acceleration 10 $\frac{m}{s^{2}}$ .

The dependence of the period of vibration of the weight suspended on the spring from its mass is expressed by the formula:

T = 2 \cdot π \cdot \sqrt{\frac{m}{k}}

where:
T - vibration period [s],
m - body weight [kg],
k - spring elasticity coefficient $[\frac{N}{kg}]$ .

Selected words and expressions used in the lesson plan

mathematical pendulummathematical pendulummathematical pendulum

weightweightweight

springspringspring

vibration periodvibration periodvibration period

frequency of vibrationsfrequency of vibrationsfrequency of vibrations

gravitational accelerationgravitational accelerationgravitational acceleration

gross errorgross errorgross error

support standsupport standsupport stand

confirmconfirmconfirm

threadthreadthread

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frequency of vibrations1

mathematical pendulum1

spring1

vibration period1

weight1

thread1

gross error1

support stand1

gravitational acceleration1

confirm1

Scenariusz

Topicmff10d7c099ed20f3_1528449000663_0Topic

Lesson stages

Selected words and expressions used in the lesson plan