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

RyL61FbBBVpMt

Wyznaczanie okresu i częstotliwości drgań wahadła matematycznego i ciężarka na sprężynie

You will learn

measure the period of vibration of the mathematical pendulum and the weight on the spring,
determine the factors affecting the measurement uncertainty and the ways to reduce this uncertainty,
study what physical quantities does the vibration periodvibration periodvibration period of the mathematical pendulum and the weight on the spring depend on.

R1DIN7lM5rwV6

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

Answer the following questions.

What is the mathematical pendulummathematical pendulummathematical pendulum?
What and how does the period of vibration of the mathematical pendulum depend on?
What is the springspringspring pendulum?
What and how does the period of vibrations of the weightweightweight on the springspringspring depend on?

Experiment

Experiment 1

Research problem

R16hy39rBEpCD1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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

Hypothesis

R1HOFYmSGm0rs1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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

You will need

RD0Ode5hcvYGO

Nagranie dostępne na portalu epodreczniki.pl

nagranie abstraktu

You can use the following things.

Ra3tI90zYyIVz1

Instruction

RFsMP13cDgRfT

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

1. Build a mathematical pendulummathematical pendulummathematical pendulum.

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:

R1NV0QMnUo3YA1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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

A suggested measurement table:


No.	Thread length l [cm]	Number of vibrations n	Vibration time t [s]	Vibration period T [s]

Summary

Processing of the results:

Rk24scCM3DESz1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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

Experiment 2

Research problem

RenEtjsPIF8rs1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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

Hypothesis

Rt6mT42XrntkO

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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

Instruction

RKHEsFym6lTPt1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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

RCl5Y1ODCJHRe1

Na rysunku przedstawione jest wahadło sprężynowe. Na statywie zawieszona sprężyna, na niej zawieszone trzy małe odważniki. Na jednym z boków statywu miarka. — Source: GroMar, licencja: CC BY 3.0.

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

3. Measure the time needed to complete n the full vibrations of the weight on the spring.

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.

Proposed measurement table:


No.	Body weight [m]	Number of vibrations n	Vibration time t [s]	Vibration period T [s]

Summary

Processing of the results:

RtynLy1MyJIwA

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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 $[\frac{N}{k g}]$ .

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 $[\frac{N}{k g}]$ ,
m - body weight [kg],
g - gravitational acceleration 10 $\frac{m}{s^{2}}$ ,
x - displacement from the equilibrium position [m].

Summary

Mathematical pendulum

RMRDXAyZ5TOxe

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

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 mathematical pendulum thread 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.

WeightweightWeight on the spring

RmFVR67D7lkeD1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

nagranie abstraktu

The conclusions from the measurements are following:

the period of vibrations of the weight on the springspringspring depends on its mass,
greater mass corresponds to a higher value of the vibration periodvibration periodvibration 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).

Theoretical formulas

R1bYKK8wRACl5

Nagranie dostępne na portalu epodreczniki.pl

nagranie abstraktu

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 $[\frac{N}{k g}]$ .

Exercises

R1BDEMXYfn5X0

Exercise 1

Determine which sentence is true.

The vibration period of a mathematical pendulum is directly proportional to the length of the pendulum thread.
The vibration period of a mathematical pendulum of the same length would be different on the Mars and on the Earth.
Two identical springs were prepared. The weight was attached to the first spring and was set in the vibrating motion. Next, the second spring was serially attached to the first one. Then the period of vibrations of the weight on the spring increased.
With the help of the weight suspended on the spring, the value of the gravitational acceleration g can be determined.

Exercise 2

For one pendulum the vibration period is 1 second. Calculate how much you should shorten the length of the mathematical pendulum to make the period of its vibration equal 0,5 seconds.

Analysis

Given:

$T_{1} = 0, 5 s$

$T_{2} = 1 s$

$g = 10 \frac{m}{s^{2}}$

Unknown:

$l_{2} - l_{1} = ?$

Solution:

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

$T^{2} = 4 \cdot π^{2} \cdot \frac{l}{g}$

$g \cdot T^{2} = 4 \cdot π^{2} \cdot l$

$\frac{g \cdot T^{2}}{4 \cdot π^{2}} = l$

$l_{2} - l_{1} = \frac{g \cdot T_{2}^{2}}{4 \cdot π^{2}} - \frac{g \cdot T_{1}^{2}}{4 \cdot π^{2}} = \frac{g \cdot (T_{2}^{2} - T_{1}^{2})}{4 \cdot π^{2}}$

$π^{2} \approx 10$

$l_{2} - l_{1} = \frac{10 \frac{m}{s^{2}} \cdot ({(1 s)}^{2} - {(0, 5 s)}^{2})}{4 \cdot 10}$

$l_{2} - l_{1} = \frac{10 \frac{m}{s^{2}} \cdot 0, 75 s^{2}}{4 \cdot 10}$

$l_{2} - l_{1} = 0, 1875 m$

Answer:

$l_{2} - l_{1} = 0, 1875 m$

Exercise 3

Write in English how you can determine the gravitational accelerationgravitational accelerationgravitational acceleration using the mathematical pendulum.

Rf5XiTLLTIhFS

Exercise 4

Indicate which pairs of expressions or words are translated correctly.

wahadło matematyczne - mathematical pendulum
ciężarek - weight
sprężyna - spring
okres drgań - vibration period
statyw - thread
nić - weight

zadanie

Source: GroMar, licencja: CC BY 3.0.

R1UhR1aQhMEQp1

Match Polish terms with their English equivalents.

spring
wahadło matematyczne
okres drgań
częstotliwość drgań
frequency of vibrations
vibration period
weight
ciężarek
sprężyna
mathematical pendulum

Source: Zespół autorski Politechniki Łódzkiej, licencja: CC BY 3.0.

Glossary

mathematical pendulum

wahadło matematyczne

R1DkE23mpLXCf1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: mathematical pendulum

weight

ciężarek

RGBi3Q2FVEr4B1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: weight

spring

sprężyna

R1X8xvTHyn9bY1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: spring

vibration period

okres drgań

R1db0Ew7R9zce1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: vibration period

frequency of vibrations

częstotliwość drgań

Ret3QpxGxC9Yq

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: frequency of vibrations

gravitational acceleration

przyspieszenie ziemskie

R2UW39ftfFkZA1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: gravitational acceleration

gross error

błąd gruby

R11odnlurTjkU1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: gross error

support stand

statyw

R3UhMSr9BKRoA1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: support stand

confirm

potwierdzić

R7BIA617OzLHs1

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: confirm

thread

nić

R1Gcbzyf9EuW21

Source: GroMar, licencja: CC BY 3.0.

Nagranie dostępne na portalu epodreczniki.pl

Source: GroMar, licencja: CC BY 3.0.

wymowa w języku angielskim: thread

Keywords

mathematical pendulummathematical pendulummathematical pendulum

weightweightweight

springspringspring

vibration periodvibration periodvibration period

frequency of vibrationsfrequency of vibrationsfrequency of vibrations

Scenariusz

Summary

Mathematical pendulum

WeightweightWeight on the spring

Theoretical formulas

Exercises

Glossary

Keywords