What effect does changing the length of a pendulum string have on the period of a simple pendulum?

 

 

 

Hannah Figueroa Velazquez

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 Introduction and Background | Hypothesis |  Materials | Method | Risk  Assessment | Data and Analysis | Conclusion | Evaluation | Limitations | Further Research Suggestions | Bibliography | Related Websites | Return to research

 

 

 

 

 

 

Introduction/Background

 

Back and forth, back and forth. The consistent motion of a pendulum seemed like magic to me as a child, and it wasn’t until I joined physics in high school that I began to understand the mathematics and science behind such a mystical object. Was it the bob at the end that moved it forward or was this a simple gravity problem? Through more research, I was able to identify the various factors that impacted the motion of a pendulum, one of them being simple yet fascinating: the length of a pendulum string.

The simple pendulum, a perfect example of Newton's first law of motion[1]: the law of inertia, faces downwards at a 90 degree angle as it’s pulled down by gravity at rest.[2] When a force is applied on it, however, the object receives inertia and must stay in motion until forcibly stopped. The structure consists of a mass or “bob” hanging from a string which is attached to a pivot point that allows for the bob to be swung at any angle. Though not exact, the motion of a pendulum with small angles resembles that of simple harmonic motion which results when a restoring force is proportional to the displacement.[3] The angular displacement is the angle of which the pendulum has been drawn back before released and for this reason when displacement is zero, this position is known as the equilibrium position as seen in figure 1.

 

Simple Pendulum: Theory, Diagram, and Formula. Figure 1[4]

Furthermore, the pendulum has been of interest to physicists dating back to 1583 when Italian mathematician and astronomer, Galileo, discovered the relationship between a swinging lamp and his pulse rate.[5] The time in which the lamp took to complete a full swing remained perfectly constant, a tool that the scientist saw could be used for precise timing. In other words, he discovered the “period” of the lamp. A period in physics refers to the time it takes to do one complete cycle. While in waves this means the amount of time it takes for a particle to make a complete vibrational cycle, in a pendulum this means the amount of time it takes to complete a full swing. When noted that the pendulum was affected by length of string and independent of its amplitude, Galileo began imagining uses for the constant oscillation of a pendulum swing such as music tempo for students, taking a pulse, and for improving the accuracy of clocks as seen in the development of grandfather clocks.[6] He wanted to conclude what factors exactly changed the period and timing of a pendulum model.

Finally, it was determined that the mass of a bob at the end of a pendulum had no impact on the period as the gravitational acceleration of which the pendulum was being pulled remained constant regardless of mass added. Likewise, the angular displacement of the pendulum had no influence on period as the greater the angle, though it’s covering a greater space, there’s also a greater amount of acceleration occurring.[7]. On the other hand, changing the length of the pendulum string as Galileo witnessed altered the amount of time it took for the pendulum to complete a full swing.

The purpose of this investigation is to determine the effect a length of a pendulum string has on the period of a simple pendulum. My independent variable is the length of the string and my dependent variable is the period of the oscillation in seconds. My controls are the mass of the bob on the pendulum and the angle at which the bob is dropped.

 

Hypothesis

 

I believe that as the length of the pendulum string increases, the period will also increase in a linear manner as longer strings will make for longer times taken to complete a full swing just as shorter strings will produce more rapid movement meaning that length and time are directly proportional.

 

Materials

 

In this experiment I used a pendulum stand that my physics teacher had available in his classroom which I set on a table so that I was able to manipulate my lengths more freely. I also used a long string which I attached to a metal weight and tied to the top of the pendulum stand at different heights depending on what length I wanted. I used a metal clamp to provide more security and ensure that the string wouldn’t slip. The height of the pendulum stand was approximately 120 cm with an estimated uncertainty of about .05 cm. The metal weight at the bottom of the string was 170 grams with an uncertainty of .05 grams. I used a meter stick to measure all of my string lengths and finally I attached a protractor with tape to the top of my pendulum stand and made sure the 90 degree mark aligned with my string when it was at equilibrium as seen in the diagram below.

 

 

Figure 2

 

 

Method

 

First, I measured the length of the string needed for my first string length and marked that point with a sharpie on the string of the pendulum. I used lengths ranging from 10.2 centimeters to 82.6 centimeters increasing each time by increments of 3.8 centimeters so that my data would be consistent. Then, I tied the string on the top of the pendulum stand at the point where the sharpie mark had been drawn using a clamp to ensure that my string would stay in place and movement wouldn’t interfere with my result.. Afterward, I grabbed the pendulum weight and drew it out from equilibrium until the string aligns with the 10 degree mark on the protractor attached to the top of the pendulum stand. As soon as the string was aligned with the 10 degree mark, I released the weight at the same time I began recording on a stopwatch the time it takes for the string to make 10 full back and forth swings. Finally, I ended the stopwatch as soon as the pendulum had finished it’s 10th full oscillation and recorded the time. I did 3 trials for each string length and 20 variations of string length total so that I had enough data points to get a clear image of the correlation between string length and period of an oscillator.

 

Risk Assessment

 

There’s no risk involved in this experiment as long as the pendulum weight is tightly secured to the string nobody is standing in the swing area of the pendulum. There’s no environmental harm done by partaking in this investigation.

 

 

 

Data and Analysis

 

 

 

Data Table 1: Raw data displaying 3 trials for 20 different length variations

 

Length of Pendulum String ± .05 cm

Time For 10 Full Oscillations ± .05 s

Trial 1

Trial 2

Trial 3

10.2

6.29

6.40

6.55

14.0

7.56

7.45

7.23

17.8

8.32

8.65

8.43

21.6

9.61

9.72

9.84

25.4

10.31

10.15

10.01

29.2

11.20

10.90

10.76

33.0

11.83

11.56

11.29

36.8

12.36

12.19

11.98

40.6

13.50

13.35

13.23

44.5

14.12

13.89

13.74

48.3

14.57

14.31

14.27

52.1

15.11

15.09

14.67

55.9

14.99

15.12

15.21

59.7

15.75

15.53

15.31

63.5

16.22

16.01

15.83

67.3

16.24

16.77

16.52

71.1

17.02

17.28

17.46

74.9

17.91

17.77

17.57

78.7

18.02

18.21

17.76

82.6

18.29

18.75

18.55

Mass of Pendulum Weight: 170 ± .5 g

Height of Pendulum: 120 ± .05 cm

 

Data File: Text .:. Excel

Data table 1 displays my raw data after 20 different string lengths with 3 trials each. I calculated that because I used a meter stick with increments of .1 centimeters, an uncertainty of .05 centimeters was reasonable which is the same logic I used for my .05 uncertainty for the height of my pendulum. As for my time, I was measuring to the .01 second which is why I chose to assume an uncertainty of .05 seconds. Finally, for my pendulum weight I was measuring its weight in grams which is why I used an uncertainty of .5 g.

 

 

 

Data Table 2: Converting the raw data into averages and calculating the uncertainty

 

Length of Pendulum String ± .05 cm

Calculating The Time For 1 Full Oscillation ± .05 s

Trial 1

Trial 2

Trial 3

Average

Average/10 Swings

Uncertainty

10.2

6.29

6.40

6.55

6.41

0.641

0.013

14.0

7.56

7.45

7.23

7.41

0.741

0.017

17.8

8.32

8.65

8.43

8.47

0.847

0.017

21.6

9.61

9.72

9.84

9.72

0.972

0.012

25.4

10.31

10.15

10.01

10.16

1.016

0.015

29.2

11.20

10.90

10.76

10.95

1.095

0.022

33.0

11.83

11.56

11.29

11.56

1.156

0.027

36.8

12.36

12.19

11.98

12.18

1.218

0.019

40.6

13.50

13.35

13.23

13.36

1.336

0.014

44.5

14.12

13.89

13.74

13.92

1.392

0.019

48.3

14.57

14.31

14.27

14.38

1.438

0.015

52.1

15.11

15.09

14.67

14.96

1.496

0.022

55.9

14.99

15.12

15.21

15.11

1.511

0.011

59.7

15.75

15.53

15.31

15.53

1.553

0.022

63.5

16.22

16.01

15.83

16.02

1.602

0.019

67.3

16.24

16.77

16.52

16.51

1.651

0.027

71.1

17.02

17.28

17.46

17.25

1.725

0.022

74.9

17.91

17.77

17.57

17.75

1.775

0.017

78.7

18.02

18.21

17.76

18.00

1.800

0.023

82.6

18.29

18.75

18.55

18.53

1.853

0.023

Mass of Pendulum Weight: 170 ± .5 g

Height of Pendulum: 120 ± .05 cm

 

 Data File: Text .:. Excel

 

 

Data table 2 shows the process I used for converting my trials into averages, turning time for 10 swings into 1, as well as measuring individual uncertainties for each one of averages. Taking the averages for each string length was simple as I just had to add up the quantities of the 3 trials and divide by 3 to get my average. For instance, for my string length of 10.2 centimeters, my calculation looked like this:

 

 

Next, I divided each of my estimated uncertainties by 10 since I calculated the time needed for the pendulum to make 10 full swings while period only refers to 1. Once I had my average periods calculated, I went through and calculated individual uncertainties which I calculated by subtracting the smallest trial value for a given string length from the highest trial value and dividing that number by 2. An example of the calculation looked like:

 

 seconds

 

Now that I had calculated my averages and uncertainties, I was able to compile all my data into a power function graph as shown below.

 

*My error bars were too small to be displayed in my graphs

Chart 

Data File: Text .:. Excel

 

 

From this graph I was quickly able to see that contrary to my hypothesis, although length of string does have an effect on the period of my pendulum, the two variables form a power function, not a linear one as I had predicted. The time period of the simple pendulum is directly proportional to the square root of its length, meaning that if I were to double the length of the string then the period would increase as well but not by exactly double. I can derive from the strong correlation and the exponent of .5, that the period is roughly equal to the square root of the length with a constant being multiplied in front. I can analyze this relationship further by linearizing my data.

Chart

 Data File: Text .:. Excel

 

Because I learned that my data could be shown as a power function, I decided to linearize it by graphing the log of my Y values against the log of my X values which resulted in a linear graph. I can use the resulting equation to deduce the value of the coefficient A, and the power n since I know that power functions are represented by  where n will be the slope of my line. Since my slope value is about .5, I can determine that to linearize my graph I will need to raise my Y value to the  power, which in this case would be  or 2.

Chart

Data File: Text .:. Excel

 

Finally, once I deduced that I needed to square my Y values in order to linearize my data, I was given a linear trendline that proved the square root relationship of length and period. Some aspects of the graph are a product of human error such as the y-intercept being 3.25E -03 implying that when the length of the pendulum string is 0, the period is not. This answer is nonsensical but certainly the intercept is close enough to 0 so that my data still remains significantly accurate and representative of the relationship. This discovery led me to wonder what were the physics behind this square root correlation between length of a pendulum string, and its period.

 

 

Conclusion:

 

 

I investigated deeper into simple harmonic motion and found a widely accepted equation for the period of a simple pendulum that only worked with small angles. With the use of geometry and different equations, I was able to derive this formula and better understand the nature of my findings.

 

Figure 3

 

 

 

In this diagram we can begin to understand the correlation between length and period of a simple pendulum as L is the length of the pendulum swing and  is the among of degrees by which the string was pulled from equilibrium. Thanks to small angle approximation, though the triangle created by the pendulum swing isn’t exactly a right angle, the angle is small enough that we assume either side could be the hypotenuse. mg is the force acting downwards on the mass which helps us to understand using trigonometry that the horizontal vector of force acting upon the mass (F) is equal to mgsin. We can calculate this since we know that:

 

  

 So in this case, the equation would look like:

 

By moving this around with algebra we can conclude that the horizontal force vector is equal to mgsin. Furthemore, we can use the triangle created by the pendulum to compare what sin equals in terms of l and x. Once again by using trigonometric rules we can identify x as opposite to and l as the hypotenuse. This leads us to the equation:

 

Through the use of substitution we can rewrite the horizontal force vector acting on the mass to be:

 

Now, we can derive the general formula for period (T) by using the formula for angular velocity (w) given by:

 

which applies for small displacements as the one used in my experiment. Next I can use the equation:

 

to help me substitute in for w which by multiplying  by the reciprocal of , I get the general of equation for period given by:

 

Finally, I can use Hooke’s Law which identifies that as long as angle of displacement is small, the restoring force of the displaced mass follows simple harmonic motion which is given by the formula of:

 

Clearly, we can see how this resembles our derived formula of:

 

where k in this case is equal to

 

Now that we’ve identified what k is equal to, we can once more use substitution to sub in for k in the equation

 

This leaves us with the equation:

 

where the mass conveniently cancels out and leaves us with our final equation for period of a pendulum:

 

 

 

 

 

Now that we have our derived formula, we can determine how accurate our results were and what disparities were caused by factors such as human error as well as external forces such as air friction. By moving around the formula, we get that the period in seconds is equal to a constant multiplied by  where the constant is . If we use 981 as our value for gravity, then this constant would be roughly equal to .2006. Comparing this to the value we got in our first graph’s equation which depicts period v length of pendulum string, you can see that the square root of length is being multiplied by a constant of .1990 which is extremely close to the constant in our derived formula with only a .0016 value difference. Likewise, once our graph has been linearized we are presented with the a slope that tells us the constant being multiplied by LA in order to get the period squared which by moving around our original equation would give us a constant of . By calculating this you get approximately .04024. When compared to the slope we got which was a value of .0414, you see that once again though slightly skewed due to error the values are strikingly similar with a difference of only .00116. Finally, we can attribute some of the calculated error to the slight disparity in value that comes with using the small angle approximation which states that when  is small enough, the sin of  approximately equal to . Looking back at our general equation for period, we see that:

 

which we can rewrite to display:

 

 

 

By moving this equation around algebraically we are left with:

 

 

Since I used an angle of 10 degrees in my investigation, that means that if I convert 10 degrees to radians I am given .1745 radians. Sin of .1745 radians is approximately .1736. This means that we can rewrite our formula to look like:

 

 

By simplifying the equation we are left with:

 

 

Meaning that the period calculated will be about .26% too high by using the small angle approximation. Of course, most of the error can be put upon human error as there were various aspects of uncertainty throughout the investigation.

Ultimately, the square root correlation is no coincidence, and you can see how the underlying reason is because the time taken to make a full swing is dependent on the value of the restoring force, or the mgsin, which in the case of small angles, we can substitute the degree for , meaning angle of displacement has virtually no effect on the period while length does. In the same way, because we can end up canceling our mass in our equation, the mass of the pendulum weight doesn’t impact the period either. The smaller length is, the greater the restoring force and therefore, the time taken to complete a full oscillation is less.

 

Evaluation

 

Finally, my results confirmed parts of my hypothesis and rejected others. I imagined that as the length of the string increased, the period would increase as well, but I wrongly predicted that the relationship would be directly proportional and result in a linear graph. Rather, I found through graphing my data and deriving the formula for the period of a simple pendulum, that the period of a function is equal to the square root of length multiplied by a constant. As you increase the length of the pendulum string, you’re decreasing the amount of restoring force as the horizontal gravitational component will have a less steep slope downward resulting in less of a force.

 

Limitations

 

Though my data was extremely similar to what one would expect given the established formula for the period of a function, there were several sources of human error that, if fixed, could provide more accurate results. Much of the error has to do with the design, as I was timing the amount taken for 10 complete oscillations with the human eye, and much of the uncertainty was due to me not being able to precisely end my stopwatch exactly at the time that the 10 swings ended. One solution for this would be to let the pendulum swing for more than 10 swings, as it would give a clearer result, but even this wouldn’t completely solve the issue of the human eye not being fast enough to clearly tell when the pendulum has reached its last oscillation. If I were to redo this experiment, I’d use technology to make my results more accurate. If I were to videotape the pendulum on a video analysis app I would be able to perfect my timings and have less margin of error in my times. I could even use slow motion features to tell when exactly the oscillation ended.

Additionally, another source of potential error was with measuring the length of my pendulum strings. Although I measured the string each time as accurately as I could, I wasn’t able to perfectly account for the amount of string that would be used to tie the string on the pendulum stand. If I were to change this, I would, instead of tying the string, use a stronger clamp that would be able to hold the weight of the pendulum which would therefore mitigate the amount of string being tied around the metal rod.

Finally, one of my greatest sources of error in the experiment would be how I struggled to accurately time and calculate the time taken for 10 complete swings as the length got increasingly shorter. I quickly noticed that although 10 degrees was a good angle for long string length, such a small displacement from equilibrium for small string lengths made the movement too rapid and shaky to properly measure. Though it’s crucial that the angle remain small in order to be able to use small angle approximation, I think for future experiments I would start my string lengths out greater so that my range would start at a greater value and I wouldn’t have to rely on inaccurate results for my first few data points.

 

Further Research Suggestions

 

As I was fascinated by the results of my experiment, I’d be curious to investigate less widely researched aspects of the simple pendulum. For instance, as I was researching I found that though there hasn't been much research done on the subject, it’s been proven that diameter of the string can actually impact how quickly the pendulum’s oscillations dampen. In the future I’d be interested to learn more about what affects dampening in a simple pendulum and how we can manipulate the speed of dampening by changing the strings being used.

 

 

 

 

 

 

 

 

 

 

 

Bibliography:

 

Admin. “Simple Pendulum Derivation of Expression for Its Time Period - w3schools.” W3spoint, 29 Feb. 2020, https://www.w3schools.blog/simple-pendulum-derivation-of-expression-for-its-time-period.

 

Buddies, Science. “Swinging with a Pendulum.” Scientific American, Scientific American, 23 Feb. 2012, https://www.scientificamerican.com/article/bring-science-home-swinging-pendulum/.

 

“The Galileo Project: Science: Pendulum Clock.” The Galileo Project | Science | Pendulum Clock, http://galileo.rice.edu/sci/instruments/pendulum.html.

 

“Gary Garber's Blog.” Gary Garbers Blog SHM in a Pendulum Comments, http://blogs.bu.edu/ggarber/interlace/pendulum/shm-in-a-pendulum/#:~:text=The%20motion%20of%20a%20simple,Law%20when%20applied%20to%20springs.&text=Where%20F%20is%20the%20restoring,and%20x%20is%20the%20displacement.

 

“Newton's Laws of Motion.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., https://www.britannica.com/science/Newtons-laws-of-motion.

 

Oscillations - University of Michigan. http://www-personal.umd.umich.edu/~jameshet/IntroLabs/IntroLabDocuments/150-11%20Oscillations[2]/Oscillations[2]%205.0.pdf.

 

Perdue, Matthew. “The Factors That Might Affect the Period of Oscillation.” Sciencing, 13 Dec. 2020, https://sciencing.com/factors-might-affect-period-oscillation-8437461.html.

 

Reinbold, Joan. “Why Does a Pendulum Swing?” Sciencing, 2 Mar. 2019, https://sciencing.com/pendulum-swing-5280650.html.

 

“Swinging on a String - Lesson.” TeachEngineering.org, 22 Feb. 2022, https://www.teachengineering.org/lessons/view/cub_mechanics_lesson09. \

 

 

Related Websites:

https://www.acs.psu.edu/drussell/Demos/Pendulum/Pendulum.html

This site is helpful in explaining the derivation of the formula for period of a simple pendulum as well as displays diagrams that allow for one to see the forces acting on the bob as it swings.

http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html

This site allows for you to insert length of string and therefore calculates period for you. Your data should be a little different but similar meaning this is a good site to check that you’re on the right track.

 

https://file.scirp.org/Html/7-1720775_73856.htm

This site explores a different but similar aspect which is the effect of mass on string on damping of a simple pendulum. It’s more advanced but a good potential project for research.

 

https://openstax.org/books/physics/pages/5-5-simple-harmonic-motion#:~:text=A%20pendulum%20in%20simple%20harmonic,pendulum%20is%20hanging%20straight%20down).

This site is useful because it explains simple harmonic motion which is necessary for your introduction and background if you’re doing anything related to pendulums.

 

https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/harmonic-motion/v/pendulum

This is a video that I watched to get a better understand of SHM and how to draw my diagrams when explainingf the forces acting on the pendulum.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



[1] https://www.britannica.com/science/Newtons-laws-of-motion

[2] https://sciencing.com/pendulum-swing-5280650.html

[3]http://blogs.bu.edu/ggarber/interlace/pendulum/shm-in-a-pendulum/#:~:text=The%20motion%20of%20a%20simple,Law%20when%20applied%20to%20springs.&text=Where%20F%20is%20the%20restoring,and%20x%20is%20the%20displacement

[4] https://www.sciencefacts.net/simple-pendulum.html

[5] https://www.teachengineering.org/lessons/view/cub_mechanics_lesson09

[6] http://galileo.rice.edu/sci/instruments/pendulum.html

[7] https://sciencing.com/factors-might-affect-period-oscillation-8437461.html