Companion to the reading passage Galileo's Bells: Timing Gravity with a Tilted Ramp and a Musician's Ear. Answer each question, then click Check. Multiple-choice and number questions grade themselves; for written answers, compare yours with the model answer and checklist, then score yourself honestly.
Around 1604, Galileo Galilei wanted to study falling objects, but a dropped ball falls too quickly to time with the clocks of his day. He slowed the motion by letting a bronze ball roll from rest down a long, smooth, gently tilted ramp. To time the ball, he hung small bells along the ramp and listened as the ball struck them, comparing the rings to a steady musical beat.
A group of students recreated the investigation. They released a ball from rest at the top of a long ramp and moved bells along the ramp until each bell rang exactly in step with a metronome that clicked every 0.50 s. Their results are shown in Table 1.
Table 1: Bell Positions for a Steady Beat
| Bell | Time After Release, t (s) | Distance from Release Point, d (m) |
|---|---|---|
| 1 | 0.50 | 0.05 |
| 2 | 1.00 | 0.20 |
| 3 | 1.50 | 0.45 |
| 4 | 2.00 | 0.80 |
| 5 | 2.50 | 1.25 |
Data are representative values for practice.
1. Which graph best represents the relationship between the ball’s distance from the release point and time?
2. Substitute numerical values into a mathematical representation that can be used to determine the magnitude of the ball’s acceleration on the ramp. Use the data for bell 5. Then calculate the acceleration.
Next, the students placed four bells at equal spacing, 0.20 m apart, along the same ramp. They again released the ball from rest at the top and used a video recording to find the time at which each bell rang. Instead of a steady beat, the rings came faster and faster. Their results are shown in Table 2.
Table 2: Equally Spaced Bells
| Bell | Distance from Release Point, d (m) | Time After Release, t (s) | Time Since Previous Ring (s) |
|---|---|---|---|
| 1 | 0.20 | 1.00 | 1.00 |
| 2 | 0.40 | 1.41 | 0.41 |
| 3 | 0.60 | 1.73 | 0.32 |
| 4 | 0.80 | 2.00 | 0.27 |
Data are representative values for practice.
3. A student makes the following claim.
Place a check mark in either the Support or Refute box. Justify your response using a mathematical relationship.
4. Which statement best explains why the time between rings decreases in Table 2?
Galileo argued that a ramp is a model of free fall: as a ramp is made steeper, the ball’s acceleration increases, and a vertical “ramp” is the same as simply dropping the ball. On a shallow ramp the ball speeds up in the same pattern as a falling ball, but slowly enough that the intervals between bells can be judged by ear.
Consider the Earth–ball system as the ball rolls from rest at the top of the ramp to the bottom. Assume energy losses due to friction and air resistance are negligible. Take the gravitational potential energy of the system to be zero when the ball is at the bottom of the ramp.
5. Construct an energy bar graph for the Earth–ball system at the moment the ball has descended half of the ramp’s total vertical height. Show each type of energy as a fraction of the system’s total energy, ET. (PEg = 0 at the bottom of the ramp; friction and air resistance negligible.)
Click a box to set each bar’s height; click it again to clear the bar to 0. The ET bar is drawn for you.
6. Which statement best describes an advantage of using a shallow ramp instead of a vertical drop in Galileo’s investigation?