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).
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:
Author:
High school
FizziQ Web
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.