Pendulum Motion

Contributed by Siddharth Mukherjee

 How Pendulums Are Waves!

Introduction

• When you first think of something that’s a wave, something like sound waves or waves in the ocean tend to come to mind. However, in physics, it’s possible to represent any system which has repetitive motion as a wave. In this experiment, we’ll be showing how a simple pendulum can be represented as a wave.

Materials

• 1 piece of string or floss
• 1 marble, rock, or other small spherical object
• 1 piece of white paper(graph paper also works)
• 1 ruler(if you don’t have graph paper)
• A phone camera to record the pendulum
• 5 pushpins to put the pendulum and paper up on a wall

Procedure

1. Put together your pendulum. You can tie and knot your string or floss around whatever object you choose to make your pendulum. Using tape to connect the string and object is also a possible option.
2. Draw regular lines along your white piece of paper, so that the result is a grid with squares of whatever length you decided. In my case, I made my squares 2 cm long. Also make sure to label the lengths of each square for easier measuring later. Right now, your system should look like the image to the right:
3. While recording(in slow-motion mode if possible), move the pendulum up to whichever side you want and let go of it. Make sure to record how the pendulum moves and make sure your measurement grid is good enough to accurately understand what it’s position is.
4. Take the video and use it to make a table with measurements of where the pendulum is after each second or half-second. It should have 3 columns, one for time, one for x-position, and one for y-position. In my case, my table looked like this:
   

Time	X-Position	Y-Position
0	10	1
1	-9	0
2	8	0
3	-9	0
4	8	0
5	-6	-1
6	7	-1
7	-2	-2
8	3	-2

5. Plot the time and x-position values on a graph, where the horizontal axis is time and the vertical axis is the x-position. After that, you can connect the points with lines. For the data points I got, the plot should look like this:

Physics Concepts and Questions

• When looking at time rather than position, the back-and-forth motion of a pendulum causes it to go up and down in a wavelike motion.
• The waves also seem to get smaller and smaller the more time goes on, just like how sound waves get weaker the more time goes by after they’re created due to getting more spread out.
• It turns out that due to gravity, the way that a pendulum goes back and forth can actually be represented by a smooth wave. This is due to the way that gravity pulls down on the marble while the string resists it’s motion. 
• If you actually plotted out an infinite amount of points for a pendulum, the final result would look something like this. You can try measuring more points and seeing if you can make your measurements look like this ideal result.

Conclusions

• In summary, we can see that the description of a wave is not limited to things that you may generally consider to be waves, like sound or ocean waves. 
• If we plot out any object with moves back and forth with respect to time, we can represent that object as a wave. Since a pendulum moves back and forth, that makes its movements in the x-direction and y-direction waves. 
• This concept in physics is called a parametric function, and is one of the most useful tools that physicists use in nearly every form of physics. 

Further Investigations

• You can try seeing the graph for the y-axis and seeing how it differs from the graphs for x. If you do it right, you should see peaks in different places from the time vs. x-position graph.
• You can also try taking measurements every 0.25 seconds and seeing how the graph becomes more smooth. If you do it right, you should see the result becoming more like a smooth wave that slowly gets smaller. 
• You can try changing various other parameters of the pendulum, like the mass on the string, the length of the string, or even the shape of the pendulum object. You should find that only changing the length of the string changes the time it takes for the pendulum to go from one side to the other.