# Pendulum

## Purpose

Less frequently seen today, a grandfather clock's image is iconic– the large face accentuated by its long and rhythmic pendulum. This pendulum is an important element in keeping time and is precisely engineered to fill this role. Keeping accurate time is such an important task that there are entire divisions at national laboratories responsible for doing so. While those departments utilize different techniques, investigating the harmonic motion of a pendulum can help develop an understanding of the role they fill in keeping time.

In this lab, you are tasked with developing a model describing the relationship between the parameters affecting a pendulum and its period. Starting with the simplest model and assumptions (a string), you can build upon this understanding in order to develop a model for different pendulums, including those of rigid bodies.

## Theory

A (relatively) heavy mass on the end of a string is one of the simplest pendula. You are being tasked with modeling this system; we will not provide the equations that are typically used. While they are easily found online, it is important to recognize there are assumptions that go into those models. In this lab, it is likely that your data and analysis will not fit those simple models well.It will be your job to develop these models and discuss how you know when they are applicable.

## Research Concepts

In order to successfully investigate pendulum motion in lab,it may be helpful to come in with an understanding of:

- Simple harmonic motion

- How to best plot your data in Excel in order to develop model

- The importance of timekeeping scientifically (https://goo.gl/Em2fX6)

## Impulse

**Part 1a – Pendulum Observations**

At your station, you have everything you need to create a simple pendulum-‐ a mass on a string. Create a simple pendulum and set it in motion. Explore its motion and make observations regarding properties you find interesting.

- Where is the bob moving the fastest? The slowest?

- Can you make modifications to adjust the speed of the pendulum?

- Can you make modifications to adjust the period of the pendulum?

- Is its speed and period related and, if so, how?

Take note of the parameters you are adjusting, as you will model how the period depends-‐ and doesn't depend -‐ on them in the next section?

**Part 1b – Systematic Decisions**

As you try to model the system of a pendulum, it is important to ensure you are taking your data systematically. There are many parameters that may affect your system and there may be choices you make where the model doesn't apply,so exploring what may cause you to draw false conclusions can help you develop an experimental design that probes what is important. In addition to any parameters that you decide are important, it would be useful to explore the differences in period the longer the pendulum swings. In doing so, consider:

- Does the period change the longer the pendulum remains in motion?

- Is there a certain oscillation that is more valuable to measure than others (i.e., is the first oscillation more useful than the fifth, tenth, or hundredth)?

- How will these conclusions help you determine how you will obtain a value for the period?

**Part 2 – Modeling the Period**

You and your group have likely noticed a number of adjustable parameters that have an effect on the motion of the pendulum. Your group is tasked with modeling the relationship between these parameters and the period of the swing. It will be up to your group to determine the experimental design of these measurements, but it may be useful to consider:

- What parameters does you group this is relevant? Do you have any expectations about what adjusting them will do?

- What parameters are you adjusting each time, and what are you holding constant?

- How are you measuring these parameters for each trial? (ex. How are you measuring the time of the period -‐ stopwatch, photogate, video tracker, other?)

- What are some sources of uncertainty for these measurements? Can you design your experiment to minimize the potential uncertainty?

- Can you define your minimum-‐difference threshold when determine if something affects the period?

- How will you visually represent these data to demonstrate these relationships?

**Part 3 – A Rigid Pendulum**

So far, you have considered the string of the pendulum to be massless, a relatively appropriate approximation. Switch the string for a rigid body (meter stick) and measure its period.

- What is the difference between the string and the rigid body? What assumptions have gone into the string model that are no longer applicable?

- What length of string for a pendulum in Part 2 would result in the same period?

- Can you construct and experiment to investigate an appropriate model for the period of a rigid-‐body pendulum?

**Part 4 – Challenges**

**Option A** – Pushing the Limits

You may have noticed that some of your models were not consistent. That is,your model only worked in certain circumstances. Test those limits to determine:

- Which parameters work in all circumstances, and which only work for certain situations?

- What are these situations that alter the model?

- Can you determine exactly the limit?

- Can you determine the reason this limit exists?

**Option B** – Testing the Rigid-‐Body Model

Like you did in Part 2 , develop a model for a pendulum using a rigid body. How does this compare to your prediction in Part 3?

## Questions to Think About

While conducting the experiment, consider the following questions:

- How many data points are worth taking to obtain a clear relationship between parameters?

- How are you determining a reasonable fit of your data?

- What are sources of uncertainty that arise?

- How can you minimize the effect of these sources?

- How can you consider these sources when analyzing your data?