# Frequency and the Pendulum

Overview:

## Pendulum Activity! Frequency, Length, and Gravity

*Image Credit: http://blog.echovar.com/archives/2580*

A **Pendulum **is a weight (bob) that swings back and forth.

**Frequency **is how many times something happens in a certain amount of time—like how many times a pendulum swings back and forth in 30 seconds.

**Materials**

-Pencils

-Note Pad or graph paper

-Ruler

-String (something that isn't stretchy, so it doesn't lose energy to friction)

-A really heavy book or a clamp

-Three different small weights (I simply used washers from the hardware store. For length one I used one washer, length two I used two washers, and length three I used three washers): The mass is the weighted part of the pendulum that swings in a constant motion when released. Any item that has some weight but not enough to break the string should work.

-Stopwatch

-Spaceship to test your pendulum on different planets (just kidding)

**In this activity you’ll figure out how the pendulum’s frequency depends on the length of the string, the weight of the bob, and the force of gravity. We can’t change the force of gravity (not without a spaceship) but we can vary the weight and the length, and see how that affects the frequency.**

**SET UP:**

Step 1: Cut three different lengths of string and tie the different weights to the ends. Try increasing the lengths by a factor of four (e.g. 10cm, 40 cm, 160cm). Cut three pieces of string at each length and tie the three different weights to each.

Step 2: Place a ruler on the edge of a table so that it sticks out about 10cm/4 inches

Step 3: Put a really heavy book or clamp on the end of the ruler that is on the table and tie your first pendulum to the ruler

Step 4: Prepare the following table:

Step 5: follow the steps below and do the 3 different trials

**Trial one: three 10cm lengths of string with Mass 1 (1 washer), Mass 2 (2 washers), and Mass 3 (3washers):**

Step 1: Tie one end of each of the three 10cm strings to each mass

Step 2: Tie your Mass 1 string to the ruler and get your stopwatch ready for 30 seconds!

Step 3: Ready? Set? Go! Now count the number of times the pendulum swings back and forth in 30 seconds

Step 4: Record your results in the table

Step 5: Now repeat steps 1-4 with Mass 2 and 3

** **

**Trial two: three 40cm lengths of string with Mass 1, Mass 2, and Mass 3:**

Step 1: Tie one end of each of the three 40cm strings to each mass

Step 2: Tie Mass 1 to the ruler and get your stopwatch ready for 30 seconds!

Step 3: Ready? Set? Go! Now count the number of times the pendulum swings back and forth in 30 seconds

Step 4: Record your results in the table

Step 5: Now repeat steps 1-4 with Mass 2 and 3

** **

**Trial three: three 160cm lengths of string with Mass 1, Mass 2, and Mass 3:**

Step 1: Tie one end of each of the three 160cm strings to each mass

Step 2: Tie Mass 1 to the ruler, now get your stopwatch ready for 30 seconds! (Note: you'll have to find somewhere high up to swing the pendulum, like a chair on a table or a tall ledge)

Step 3: Ready? Set? Go! Now count the number of times the pendulum swings back and forth in 30 seconds

Step 4: Record your results in the table

Step 5: Now repeat steps 1-4 with Mass 2 and 3

**Observe the Variables: Analysis Level One (easy):**

Look at your table, and discuss how changing the mass and length affected the frequency. Does the mass matter? Does the length matter? Does increasing them increase or decrease the frequency?

**Let's do the Math! Analysis Level Two (more difficult):**

Now that we have some data, let’s do some theory, and see if it fits!

What we want is a formula that gives the frequency as a function of length, time, and the force of gravity. This may sound like a hard problem, but there’s a clever shortcut that physicists and mathematicians use all the time. It has to do with the units we use to measure things.

There are three basic kinds of units: length, time, and mass. A speed is length divided by time, like “60 miles per hour”. So its units are **length/time**. If you want to know how far you get in a certain amount of time, you multiply:

**speed*time = (length/time)*time = length **

In other words, if the only ingredients you have are a speed **s** and a time** t**, and you want a length **l**, the only possible way to get it is

**l=s*t**

An acceleration is length divided by time, and divided by time again: for instance, if a car goes from 0 to 60 in 6 seconds, it accelerates at 10 miles per hour per second. So its units are

**(length/time)/time = length/time ^{2}**

The acceleration of gravity, which we call **g** has the same units: it’s 9.8 meters per second per second.

Now suppose you drop a rock off a bridge. How long will it take to hit the water? Your only ingredients are **g** and the height **h** of the bridge:

**g = length/time ^{2}**

**h = length**

and you want a time **t**. How are you going to get it? You might start by dividing **h** by **g**: this gives units

**h/g = length/(length/time ^{2}) = time^{2}**

This is a **time ^{2}**, not a time. So we can take the square root, and write

** **

** **

This is almost right. The exact answer is But even though we didn’t get that **2**, we got the right scaling—we know with what multiplications, divisions, and powers **t** depends on **h** and **g**. People write this like** **

Once we know this, we know by what ratio **t** will change if we change **h** or **g**. For instance, if we multiply **h** by **4**, then **t** gets doubled, since

** **

That is, it takes twice as long to fall four times as far. And if we move to a planet where gravity is **4** times as strong as on Earth, **t** will get divided by **2**:

** **

On that planet, it takes half as much time to fall the same distance as on the Earth.

The fancy word for all this is dimensional analysis. Congratulations! Now you know how to solve lots of physics problems!

Now back to the pendulum. You have just three ingredients:

**. the length L of the pendulum**

**. the mass M of the bob**

**. the acceleration g of gravity**

Your job is to combine these in a way that gives a frequency **f**. The units of frequency are **1/time**, as in “swings per second”: the number of swings is just a number, so its units are **1**.

There’s just one way to combine **L**, **M**, and **g** that will give you** f** with units **1/time**. Figure out how!