Introduction

When Galileo introduced the concept of uniform acceleration, he defined it as equal increases in speed in equal intervals of time. This experiment is similar to the one discussed by Galileo in his book, Dialogues Concerning Two New Sciences, in which he assumed that a ball rolling down an incline accelerates uniformly. Rather than using a water clock to measure time, as Galileo did, you will use a Motion Detector connected to a computer. This makes it possible to very accurately measure the motion of a ball rolling down an incline. From these measurements, you should be able to decide for yourself whether Galileo’s assumption was valid or not.

Galileo further argued in his book that balls of different sizes and weights would accelerate at the same rate down a given incline or when in free fall. This was contrary to the commonly held belief of the time that heavier objects fall at a greater rate than lighter objects.

Since speed was difficult for Galileo to measure, he used two quantities that were easier to measure: total distance traveled and elapsed time. However, using a Motion Detector it is possible to measure much smaller increments of time, and therefore calculate the speed at many points down the incline. The data you will be able to gather in one roll of a ball down an incline, is more than Galileo was able to acquire in many trials.

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Prelab

1. List some observations that led people of Galileo’s time to believe that heavier objects fall faster than lighter objects. Give some real thought as to why people would think this.

2. Drop a small and a large object from the same height at the same time. Very carefully determine if the larger one hit first, last, or at the same time? Describe what you actually saw.

3. Now try again simultaneously dropping two objects, but this time start the small object about 30 cm above the larger one.
        - Based on what you observe, does the distance between the two objects increase, decrease, or remain the same as they fall?
        - What is expected to happen? Explain using Galileo’s assumption of constant acceleration.

With dropping objects, the time is very short and so it's hard to tell just what happens by eye. If you do these simple experiments critically, you may see why Galileo and the people of his day had a difficult time answering the questions of motion.

Extra (no mark for this): Guess as to what will happen when rolling balls of different size and mass down an incline. Will they all have the same acceleration? Explain why you think this.

Procedure

1. Check the connection of the Motion Detector to the LabPro box and to the computer. Make note of which ports are used.

2. The Motion Detector is clamped at the top end of the incline. Determine the angle of the incline using trigonometry with measurements of some lengths/distances.

3. Prepare the computer for data collection by opening the LoggerPro software with the "MotionGalileo" experiment. Notice what is being plotted on the graphs and the scaling being used. Scaling of the graphs displayed can be adjusted during the experiment by double clicking and changing the end number on the scale. Adjust the distance scale to roughly the length of your channel.

4. Position a ball about ½ m down the incline from the Motion Detector. and press "Collect" to begin data collection. Release the ball when you hear the Motion Detector start to click. Do this a few times until you have a good representative view into this experiment.

5. Scale your two graphs for the best view of the ball rolling region. Adjust the x-axis scale to be the same for both graphs with the y-axis of each graph adjusted for the best/largest view of the data for that graph. Record in your notebook (by sketching and including important details) the graphs of distance vs. time and velocity vs. time. Identify the region on your graph that corresponds to the ball rolling down the incline.

6. What algebraic curve does the distance vs. time graph appear to follow?
   Try fitting various functions to the portion of the data corresponding to ball rolling region by dragging across that time interval and selecting "Analysis, Automatic Curve Fit" from the menu. Select a function from the scrolling list and click "Try Fit" button. Try several functions leading to the most simple function that fits well.
   Select the function that relates to uniform acceleration theory (a quadratic) for fitting to your data. Record both the equation and the parameters of this fitted equation. Record in your notes a sketch of the graph with the curve fit information.

7. Similarly, does the velocity vs. time graph follow a simple algebraic curve? Using the process above, choose the simplest function that still fits the data well and record the parameters of the fitted equation. Record in your notes the graph with the curve fit information.

Analysis

1. For the region on your graph where the ball is rolling down the incline, record in a data table the value of time and velocity for point where the ball begins to roll, by reading the data value from your graph. Then starting from that point, record time and velocity data for equal time increments (every 0.1s or 0.2s) up to the end of the acceleration such that you have at least 10 data points,.

2. Calculate the change in speed between each of the points in your data table above. Enter these values in the right column of the data table.

3. As stated earlier, Galileo’s definition of uniform acceleration is equal increases in speed in equal intervals of time. Does your data support or refute this definition for the motion of an object on an incline? Explain.

4. Was Galileo’s assumption of constant acceleration for motion down an incline valid? How do your data support your answer?

5. Calculate the average acceleration of the ball between the first and the last time recorded (t₁ and t_last) using your data and the definition of average acceleration:

Data and Result Table

Procedure Part 2

1. Determine the average acceleration for at least 1 other types of ball (of different size or mass) and for a ("Hot Wheel") cart. Summarize in a table all the accelerations found with the balls/carts used. Include a rough measurement of size and mass as well as description.

Notice the accelerations are different. Which had the greatest and which had the least acceleration? Can you hypothesize as to why the accelerations are different? Does what you found here not contradict what Galileo stated?

2. Roll a ball up the incline so that it slows to a stop at the top and then rolls back down without coming within a half meter of the Motion Detector. Compare the up and down motion to the down only motion that you examined in the lab. Is the acceleration the same during the whole motion? Is the velocity always the same sign? How could you change the coordinate system of your experiment so the acceleration changes sign?