Leibniz

This activity allows students to experimentally verify the principle of conservation of energy for a pendulum. It links theoretical calculation and practical measurements to validate a fundamental law of physics.

Gottfried Wilhelm Leibniz, the great German polymath of the late 17th century, was among the first to articulate the principle that would become the conservation of energy. He introduced the concept of vis viva (living force), which we now call kinetic energy, and argued that this quantity is preserved in mechanical processes. For a pendulum, Leibniz's principle makes a powerful and testable prediction: at the lowest point of the swing, all potential energy has been converted into kinetic energy, and the centripetal acceleration at that point is directly determined by the starting height. By combining the energy conservation equation (mgh = ½mv²) with the centripetal acceleration formula (a = v²/r), we obtain a remarkably simple relationship: the maximum centripetal acceleration at the bottom of the swing equals 2gh/r, where h is the release height and r is the pendulum length. This experiment tests this prediction directly, using a smartphone as the pendulum bob and its built-in accelerometer to measure the centripetal acceleration at the lowest point.

Learning objectives:

The student studies the conservation of mechanical energy of a pendulum using the FizziQ accelerometer. After having established by calculation the theoretical relationship between the initial height and the maximum centripetal acceleration at the low point, the student carries out experimental measurements for different heights and verifies graphically that the centripetal acceleration is indeed proportional to the height.

Level:

High school

FizziQ

Author:

Duration (minutes) :

40

What students will do :

- Derive the theoretical relationship between release height and maximum centripetal acceleration for a pendulum
- Measure the centripetal acceleration at the lowest point of a pendulum swing using the smartphone accelerometer
- Verify the conservation of mechanical energy by comparing theoretical and measured accelerations
- Investigate how the release height affects the maximum acceleration
- Understand the connection between potential energy, kinetic energy, and centripetal acceleration

Scientific concepts:

- Energy conservation
- Simple pendulum
- Centripetal acceleration
- Potential and kinetic energy
- Periodic movement

Sensors:

- Accelerometer (centripetal/radial acceleration component)

What is required:

- Smartphone with the FizziQ application
- Strong string or wire to create a pendulum
- Material for fixing the pendulum
- Tape measure
- Support for measuring different heights
- FizziQ experience notebook

Experimental procedure:

Construct a simple pendulum by attaching the smartphone securely to a strong string or wire, at least 1 meter long. The phone should be firmly fixed so it does not swing independently of the string.
Measure the length of the pendulum L from the pivot point to the center of the smartphone. Record this value.
Open FizziQ and select the Accelerometer sensor. Choose the axis along the string direction (radial direction), which will measure the centripetal acceleration.
Before the first trial, calculate the theoretical prediction: for a release height h, the maximum centripetal acceleration at the bottom should be a_max = g + 2gh/L (including gravity).
Pull the pendulum to a small angle corresponding to a known height h₁ (measure h from the lowest point to the release position). Start with h₁ ≈ 5 cm.
Start recording in FizziQ, then release the pendulum from rest. Let it swing for 3-4 complete oscillations.
Stop recording and identify the peak acceleration values on the graph. These peaks occur at the lowest point of each swing.
Record the average peak acceleration for the first 2-3 swings (before friction significantly reduces the amplitude).
Repeat with larger release heights: h₂ ≈ 10 cm, h₃ ≈ 15 cm, h₄ ≈ 20 cm, h₅ ≈ 30 cm.
For each height, record the measured peak acceleration and the theoretical prediction.
Plot a graph of measured peak acceleration versus release height h. It should be a straight line with slope 2g/L.
Compare the measured slope with the theoretical value 2g/L and calculate the percentage discrepancy.
Discuss whether the conservation of energy is confirmed by your data and identify the sources of any discrepancy.

Expected results:

For a 1-meter pendulum, the theoretical centripetal acceleration at the lowest point (including gravity) increases linearly from about 10.8 m/s² (h = 5 cm) to about 15.7 m/s² (h = 30 cm). The measured values should follow this linear trend within 5-15% accuracy. The graph of peak acceleration versus h should be linear with a slope close to 2g/L ≈ 19.6 m/s²/m for a 1-meter pendulum. Energy losses from air resistance and string friction cause the measured acceleration to be slightly less than predicted, especially at larger amplitudes. The accelerometer itself may also show some noise and axis misalignment, contributing ±0.2-0.5 m/s² of uncertainty. Over multiple swings, the peak acceleration should decrease gradually as energy is dissipated.

Scientific questions:

- How does the principle of conservation of energy predict the centripetal acceleration at the lowest point?
- Why does the measured acceleration include a component of g even when the pendulum is at the lowest point?
- What happens to the energy that is lost during each swing? Where does it go?
- How does the pendulum length affect the relationship between height and acceleration?
- Why did Leibniz call kinetic energy vis viva (living force)?
- How could you modify this experiment to minimize energy losses?

Scientific explanations:

Gottfried Wilhelm Leibniz (1646-1716) was one of the first to formulate the principle of conservation of energy, a fundamental concept of physics. For a simple pendulum, this principle establishes that the total mechanical energy remains constant in the absence of friction.

This conservation makes it possible to establish a direct relationship between the starting height and the centripetal acceleration at the lowest point. The theoretical analysis proceeds as follows: 1) At the starting point, at a height h, the energy is entirely potential: E_p = mgh (m: mass, g: gravitational acceleration); 2) At the lowest point, the energy is entirely kinetic: E_c = ½mv² (v: tangential velocity); 3) By conservation of energy: mgh = ½mv², therefore v = √(2gh); 4) The centripetal acceleration is linked to the tangential speed by a_c = v²/r (r: length of the pendulum), which gives: a_c = 2gh/r.

This last equation shows that the centripetal acceleration is directly proportional to the initial height h, with a coefficient of proportionality 2g/r. The experiment consists of verifying this relationship by measuring the centripetal acceleration for different starting heights.

The smartphone's accelerometer, placed at the end of the pendulum, directly measures this acceleration at the lowest point of the movement. The graphical representation of a_c as a function of h should give a line whose slope allows the ratio 2g/r to be estimated experimentally.

Any deviations are mainly explained by friction (air and suspension point) which dissipates part of the energy, particularly for large amplitudes. This experiment perfectly illustrates the predictive power of the principle of conservation of energy, a pillar of classical mechanics.

Extension activities:

Frequently asked questions:

Q: Which accelerometer axis measures the centripetal acceleration?
R: The centripetal acceleration is directed along the string, toward the pivot. Depending on how you attach the phone, this may be the X, Y, or Z axis. Before starting, gently swing the pendulum and check which axis shows the largest oscillations; that is your radial axis.

Q: The peak acceleration values decrease with each successive swing. Why?
R: Energy is gradually lost to air resistance and friction at the pivot. Each swing has slightly less total energy, resulting in a lower speed at the bottom and hence a lower centripetal acceleration. Use the first 2-3 swings for the most accurate comparison with theory.

Q: My measured accelerations are consistently lower than the theoretical values. What could explain this?
R: Energy dissipation is the most likely cause. Also check whether the string stretches slightly under load, which would increase the effective pendulum length and reduce the acceleration. Verify your height measurements carefully.

Q: Can I use a short pendulum (30-40 cm) for this experiment?
R: Yes, but shorter pendulums swing faster, making it harder to identify peak values. Also, the smartphone's mass becomes a larger fraction of the total and its size may not be negligible compared to the string length. Longer pendulums (≥1 m) give cleaner results.