Galileo's inclined plane
Verify Galileo's law on the inclined plane: the distance traveled is proportional to the square of time, and the acceleration equals g × sin(α).
In 1604, Galileo had a brilliant idea: to study falling objects, which were too fast to measure by eye, he 'diluted gravity' by rolling balls down an inclined plane. He discovered that the distance traveled was proportional to the square of the elapsed time. This fundamental relationship, d = ½at², became the foundation of kinematics. The FizziQ Web Inclined Plane simulation lets you reproduce Galileo's experiment and verify his law by measuring position versus time for different inclination angles, then showing that the acceleration is proportional to the sine of the angle.
Activity overview:
The student uses the FizziQ Web Inclined Plane simulation to record a ball's position versus time for different angles. They verify that distance is proportional to t² (uniformly accelerated motion) and that acceleration equals g × sin(α).
Level:
High school
FizziQ Web
Author:
Duration (minutes) :
30
What students will do :
- Experimentally verify the relationship d = ½ × a × t² on an inclined plane
- Measure the acceleration of a ball for different inclination angles
- Verify the relationship a = g × sin(α)
- Plot and interpret d versus t² graphs to extract the acceleration
- Determine the value of g from the slope of the a versus sin(α) graph
Scientific concepts:
- Uniformly accelerated rectilinear motion
- Relationship d = ½ × a × t²
- Acceleration on an inclined plane: a = g × sin(α)
- Weight component along the plane
- Proportionality and linearity
Sensors:
- FizziQ Web Inclined Plane simulation
Material needed:
- Computer, tablet, or smartphone with FizziQ Web
Experimental procedure:
Open the Inclined Plane simulation in FizziQ Web (Experiment → Simulations → Inclined Plane).
Set the inclination angle to 30° and the travel distance to maximum. Start a recording (REC) and let the ball roll to the bottom.
The position-time data are automatically exported to the experiment notebook. Observe the graph: the curve is not a straight line but a parabola.
Add a calculated column to the table: t² (time squared). Plot the graph of distance versus t². This should be a straight line through the origin.
The slope of this line equals ½ × a. Calculate the acceleration a for the 30° angle. Note the value in a summary table.
Repeat the experiment for angles 10°, 20°, 40°, 50°, and 60°. For each angle, calculate the acceleration from the slope of d(t²).
Create a summary table with three columns: Angle, sin(angle), and Acceleration.
Plot the graph of Acceleration versus sin(angle). The curve should be a straight line through the origin.
The slope of this line gives the value of g. Compare your value with 9.81 m/s².
What happens at an angle of 90°? The acceleration should equal g: that is free fall!
Expected results:
The d(t) graph is a parabolic curve, confirming non-uniform motion. The d(t²) graph is a straight line through the origin, confirming d = ½ × a × t². The acceleration increases with angle: approximately 1.7 m/s² at 10°, 4.9 m/s² at 30°, and 8.5 m/s² at 60°. The graph a(sin α) is a straight line through the origin with slope g ≈ 9.81 m/s². The mass of the ball does not affect the results.
Scientific questions:
- Why is the position-time graph a curve and not a straight line?
- What does the slope of the d(t²) graph represent?
- Why does the acceleration not depend on the mass of the ball?
- If you double the angle from 15° to 30°, does the acceleration double?
- What would change if there were friction between the ball and the plane?
- At what angle is the acceleration exactly half of g?
Scientific explanations:
On an inclined plane, the weight of the ball decomposes into two components: a component perpendicular to the plane (balanced by the normal reaction) and a component parallel to the plane that drives the ball downward: F_parallel = m × g × sin(α).
By Newton's second law (F = m × a), the ball's acceleration is a = g × sin(α). The mass m cancels out, so the acceleration is independent of mass — just as Galileo discovered.
The larger the angle, the larger sin(α), and the stronger the acceleration. At the limit, for α = 90°, sin(90°) = 1 and a = g: this is free fall.
The motion is uniformly accelerated because the acceleration is constant (the simulation does not model friction). The position follows d = ½ × a × t² and the velocity follows v = a × t.
To verify the proportionality d ∝ t², one plots d versus t²: if it is a straight line through the origin, the relationship is confirmed. The slope gives ½a, from which the acceleration is extracted.
Extension activities:
- Why is the position-time graph a curve and not a straight line?
- What does the slope of the d(t²) graph represent?
- Why does the acceleration not depend on the mass of the ball?
- If you double the angle from 15° to 30°, does the acceleration double?
- What would change if there were friction between the ball and the plane?
- At what angle is the acceleration exactly half of g?
Frequently asked questions:
Q: The d(t²) graph is not exactly a straight line.
R: Check that the ball starts from rest (zero initial velocity). If it has an initial velocity, the relationship becomes d = v₀t + ½at² and the graph of d(t²) is no longer a simple straight line.
Q: My calculated value of g is not exactly 9.81 m/s².
R: Small deviations are normal due to measurement precision in reading the graph. A value between 9.5 and 10.1 m/s² is a good result.
Q: Why did Galileo use an inclined plane instead of free fall?
R: In Galileo's time, there were no precision clocks. Free fall was too fast to measure (about 0.45 s for 1 m). The inclined plane slowed the motion by a factor of 1/sin(α), making it measurable.
Q: Does the shape or size of the ball matter?
R: In this idealized simulation without friction, no. In reality, a rolling ball has rotational inertia that reduces the linear acceleration by a factor of 5/7.