The Ring Pendulum. A Physics Exploration of Diameter and Time Period

Table of Contents

The Research Question

History

Hypothesis

Preliminary Work

Procedure

Gathered Data

Data Processing

Plotting the Graph

Processing the data for log(T) and log(D)

Data Processing

Plotting the Graph

Creating a formula for the exact relation

Data Analysis

Conclusion

Evaluation

Bibliography

Diameter and the Time Period

—A physics exploration on ring pendulum

illustration not visible in this excerpt

Image source:

http://cs.brown.edu/people/orgs/artemis/old/2014/StudentProjects/AshleyLeonardo-3/html.finalproject/index.html

The Research Question

In the course of my day to day life, I have watched many objects and systems in oscillatory motion and have been contemplating about them almost every single day. One day, as I sat on my chair, looking at the ring in my hand oscillate, I wondered why its time period was so fast. I asked myself, “Does it depend on the size of the ring?” Being an avid fan of amusement park rides, I was then compelled to relate it to thrilling rides like Disk’O and Pirate Ship. Even though the shape was not the same, my interest insisted me to make the observations for the same. What I asked myself proved out to be true. I saw that the ring being smaller in size takes lesser time and the amusement park rides being greater in size took longer time. My qualitative observations forced me to find the quantitative results. My research question thus asks “To what extent does the diameter of the ring pendulum affect the time taken to complete one oscillation at constant linear mass density?”

History

Time is a physical quantity that can be measured by using a phenomenon that is repeated at constant frequency, such as the sunrise and sunset; and phases of the moon. One can also relate this phenomenon to the oscillation of the pendulum. Throughout history various experiments have been carried out to measure this quantity.

When I looked at the antique pendulum watch at my home I pondered if it worked on the same principle. I made some research about it and I found that the concept of measuring time and such clocks have a very interesting background. Galileo's research on the properties of pendulum in 1602 is considered to a milestone in the construction of the pendulum clock. His interest in this was sparked when he looked at the back and forth motion of a lamp in the Pisa cathedral¹.

illustration not visible in this excerpt

Imagesource:https://www.flickr.com/photos/koopmanrob/3775343287/sizes/o/

Later the pendulum clock was invented by the Dutch scientist Christiaan Huygens in 1656 on the same grounds of Galileo’s research.

Hypothesis

From the pre lab activity, I noticed that time period of such a ring pendulum must depend upon its size (diameter). My research question was bolstered by an intriguing question that I had in my mind. I asked myself again “What is the exact kind of relationship between a ring’s diameter and its time period of oscillation?” Thus my aim rested on the mathematical grounds to find out the exact relation.

For it I preferred to plot a graph between time period of the ring pendulum against the diameter of the ring with an expectation for the directly proportional relation between them.

If there would not be direct proportionality between them then I preferred the second approach as given below:

I hypothesized an empirical relationship for the time period of this pendulum with the diameter of the pendulum as stated below

illustration not visible in this excerpt

In the above equation:

T is the time taken to complete one complete oscillation, in seconds; D is the diameter of the ring, in cm;

P is the constant exponent of the diameter; K is the proportionality constant.

I have raised the diameter to an unknown power, to me, as of now because I am unsure of the exact relationship between my independent variable, the diameter and the dependent variable, the timeperiod. This value of P can have any value ranging from integers to fractions.

Preliminary Work

Ring pendulum may oscillate in a number of ways. It may oscillate about a point on its circumference in its vertical plane or to and fro perpendicular to its vertical plane about the same point.

illustration not visible in this excerpt

Drawing showing possible modes of oscillations of rings

I decided to investigate about the time period for its oscillations about a point on its circumference in its vertical plane.

1. Choosing the variables and constants:

Since in my experiment I was finding out the relationship of the diameter and the time taken one complete oscillation, I took the diameter of the ring as an independent variable. According to my hypothesis the time period must depend on the diameter, although unknown of the exact proportionality, I chose the time period of oscillation as a dependent variable. I was not sure about the dependency of time period of this ring pendulum on the density of the material of the ring but it might be a hidden factor hence I decided to keep the linear mass density constant by using same material for the rings through-out my experiment.

2. Choosing the material for the wire of the ring:

I had a variety of materials to choose ranging from iron to copper for the wire of the ring. as per their availability. Materials like aluminum, brass were also available. I decided to go for Iron. The reason to choose iron was that it is highly attracted to an electromagnet, which may be helped release the ring from rest. I also chose Iron as it was easily available and was cost effective. I knew forming a perfect ring could be extremely difficult, however I chose to weld the iron wire at both the ends to form ends to form nearly a perfect ring.

3. Choosing the diameters for the rings:

I had to choose such diameters for the rings that could easily be measured and through which the experiment could be performed in an effective and efficient manner. For this I conducted a pre-lab with a ring of diameter 2.0 centimeters. I came to a conclusion that this size of the ring, was not practical for my experiment as its time period was too small to observe and get proper values. So I decided to make the starting diameter of the ring as 5.0 centimeters. I took the diameters till 21.0 centimeters with the successive increment of 2.0 centimeters. I measured these diameters through a meter scale.

I had another choice to use Vernier caliper to measure the diameter of the ring but its range was 0 to 15.0 centimeters only and I wanted to collect the data in a range from 5.0 to 21.0 centimeters. Hence it was difficult to control independent variable by using the same Vernier caliper.

illustration not visible in this excerpt

Image showing all the nine rings in ascending order

4. Choosing method to measure the time period of the oscillation:

In my experiment, when I released the ring for oscillation by using an electromagnetic assembly, it swung to and fro about its equilibrium position. The time taken for the ring to complete one oscillation is called the period of oscillation. I measured the time period manually by using a digital stopwatch. I started the stopwatch when the ring stabilized on its plane. I did so as to get a perfect plane for oscillation as in the pre-lab I noticed that the ring was not stabilized at this plane and was shifting. The instability was probably because of the nearly imperfect shape of the rings even though I made the rings almost circular. I stopped the stopwatch exactly at the moment when it completed 10 oscillations.

It was a sufficient range for the data collection for time period because in pre lab work I noticed that after it the amplitude of oscillation was small enough and it was quite difficult to collect the data with accuracy.

5. Controlling the way by which the ring is released:

I controlled the releasing of the ring by using an electromagnet. It is often observed that one cannot release the ball by hand with the same magnitude of force with controlled velocity for every trial. Thus, by using the electromagnet I released the ring at almost same initial velocity that was 0 m/s. This reduced the chance of human error and curbed this error that subsequently effects the time period.

illustration not visible in this excerpt

Image showing a ring attached to the electromagnet

[...]

¹ Helden, Al Van. "The Galileo Project."Http://galileo.rice.edu. Rice University, 1995. Web. Oct. 2015. <http://galileo.rice.edu/sci/instruments/pendulum.html>