New Zealand
Level 7 - NCEA Level 2

# Pendulums (Investigation)

Lesson

## Objectives

• To investigate radical functions in real life.
• To identify characteristics of a radical function.
• To practice with transformations of radical functions.

## Materials

• $10$10 Small bouncy balls or weights with hooks (all the same size and mass)
• Stopwatch
• Fishing wire
• $10$10 Eyelet hooks (only if you are using bouncy balls)
• Scissors
• Internet
• Paper
• Different colored pens

## Procedure

### To set up the pendulums

1. If you are using bouncy balls screw one eyelet hook into each ball.
2. Measure out 10 different lengths of fishing wire and cut them. The lengths themselves don’t matter, it only matters that they are all different. Try to vary from small to large.
3. Tie one piece of fishing wire to the eyelet hooks in the balls (or the hooks on the weights)
4. Hang each of the strings from a place where their swing will be uninhibited. Using the ceiling may be a better idea for longer pendulums.

## Investigation

1. For each pendulum that you have created first measure the length of the pendulum.
2. Pull the ball straight to one side and then release it so that the pendulum swings in a straight line.
3. Use your stopwatch to time how long it takes the pendulum to make 10 full oscillations. One full oscillation counts as the pendulum moving all the way from the far right to the far left.
4. One full oscillation would be the pendulum moving from position X to position Z.
5. Divide the time it took for the pendulum to complete 10 full oscillations by 10 to get the time it takes the pendulum to make one full oscillation (also known as the period of the pendulum).

## Questions

1. Create a graph to show the relationship between pendulum length and the time it takes for the pendulum to complete one full oscillation. Time should be on the $x$x-axis and the length of the pendulum should be on the $y$y-axis. Be sure to label both axes and title the graph.
2. Describe the relationship between length of a pendulum and the time it takes to complete one full oscillation (also known as the period of the pendulum).
3. What kind of graph would this relationship best be modeled by?
4. Use this website to look up the equation for the period (oscillation time) of a simple pendulum.
5. Use the equation to calculate the predicted period for each of the pendulums of different lengths that you created.
6. Using a different colored pen graph the equation for the period of a pendulum on the same graph as your original data.
7. How close was your data to the predictions?
8. Answer the following questions about the graph of the equation for the period of a pendulum:
• Is it increasing or decreasing? Explain why it is or isn’t in context of the problem.
• Describe in your own words the way that the function increases or decreases.
• What is the range of the function?
• What is the smallest possible value for this function? Explain why it is in the context of the problem.
• Is the function more or less steep than the function of $\sqrt{x}$x?

#### Work with a friend !

1. Do you think that your results would be the same if you used a heavier wire or string to suspend the balls? Why or why not?
2. A Grandfather Clock uses a pendulum to keep time. All pendulums for Grandfather Clocks are calibrated so that their period is 2 seconds. What length string would you need to create this pendulum using the same balls as you did in your experiment?
3. What other uses do pendulums have in real life? You may need to use the internet to answer this question.

### Outcomes

#### M7-2

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

#### 91257

Apply graphical methods in solving problems