Huygens

This activity allows students to experimentally verify the principle of conservation of mechanical energy for a pendulum. It develops the ability to analyze an oscillatory movement and to quantify energy transformations.

Christiaan Huygens, the brilliant 17th-century Dutch scientist, invented the pendulum clock in 1656, transforming timekeeping from an imprecise art into a science. His success was based on a deep understanding of the pendulum's mechanics: as the bob swings, gravitational potential energy converts smoothly into kinetic energy and back again, with the total mechanical energy remaining constant throughout the cycle. This principle of energy conservation, later formalized by Leibniz and others, is one of the most fundamental laws of physics. In an ideal pendulum with no friction, the bob would swing forever at the same amplitude, endlessly trading height for speed. In reality, air resistance and friction at the pivot slowly drain energy from the system. This experiment uses FizziQ's kinematic analysis module to track a pendulum's motion frame by frame, calculate both potential and kinetic energy at each instant, and verify that their sum remains approximately constant, bringing Huygens' observations into the modern laboratory.

Learning objectives:

The student uses the FizziQ Kinematic Analysis module to study the movement of a pendulum. By pointing out the position of the ball during its descent and analyzing the position and speed data, the student calculates the potential energy and the kinetic energy at different moments of the movement then verifies that their sum remains constant.

Level:

High school

FizziQ

Author:

Duration (minutes) :

40

What students will do :

- Track the motion of a simple pendulum using video-based kinematic analysis in FizziQ
- Calculate the gravitational potential energy (Ep = mgh) and kinetic energy (Ec = ½mv²) at each tracked position
- Verify that the total mechanical energy (Ep + Ec) remains approximately constant during the swing
- Identify and quantify energy dissipation due to friction and air resistance
- Understand the continuous conversion between potential and kinetic energy in oscillatory motion

Scientific concepts:

- Conservation of mechanical energy
- Simple pendulum
- Potential and kinetic energy
- Analysis of oscillatory movement
- Harmonic motion equations

Sensors:

- Camera (video recording for kinematic analysis)
- FizziQ Kinematics module (frame-by-frame position tracking)

What is required:

- Smartphone or tablet with the FizziQ application
- 'Pendulum' video from the FizziQ library or personal video of a simple pendulum
- Optional: spreadsheet for energy calculations
- FizziQ experience notebook

Experimental procedure:

Open FizziQ and navigate to the Kinematics module. Load the 'Pendulum' video from the FizziQ resource library, or record your own pendulum video.
Set the scale using a known reference in the video (the length of the string, or a ruler visible in the frame).
Define the origin of the coordinate system at the lowest point of the pendulum's swing (the equilibrium position), with y-axis pointing upward.
Track the position of the pendulum bob frame by frame through at least one complete swing (from one extreme to the other and back).
Collect at least 15-20 data points across the full swing for smooth energy curves.
From the position data, calculate the height h above the lowest point for each tracked position: h = y (since the origin is at the bottom).
Calculate the velocity v at each point from the position differences between consecutive frames divided by the time interval.
Compute the potential energy Ep = mgh and the kinetic energy Ec = ½mv² at each point. You may use m = 1 (arbitrary units) since we are verifying that the ratio Ec/Ep is consistent with conservation.
Calculate the total mechanical energy E_total = Ep + Ec at each point.
Plot Ep, Ec, and E_total versus position (or versus time) on the same graph in the FizziQ notebook.
Verify that E_total remains approximately constant, while Ep and Ec oscillate in antiphase (when one is maximum, the other is minimum).
Calculate the percentage variation of E_total across the swing. For a well-tracked pendulum over one swing, the variation should be less than 10-15%.

Expected results:

The potential energy should be maximum at the extremes of the swing (where the bob is highest and momentarily stationary) and zero at the lowest point. The kinetic energy should be zero at the extremes and maximum at the lowest point. These two curves should be mirror images of each other. The total mechanical energy should remain approximately constant, typically within 5-15% variation for careful tracking. The main sources of apparent energy non-conservation are velocity measurement noise (since velocity is derived from position differences, it amplifies errors) and scale calibration inaccuracies. Over multiple swings, a gradual decrease in total energy of 2-5% per swing reflects real energy dissipation from air resistance and pivot friction.

Scientific questions:

- At what point in the pendulum's swing is the kinetic energy maximum? The potential energy? Why?
- Why does the total mechanical energy remain constant in an ideal pendulum?
- What factors cause the total energy to decrease over time in a real pendulum?
- How does the period of a simple pendulum depend on its length? On the mass of the bob?
- What was Huygens' key insight that made the pendulum clock so accurate?
- How would the energy analysis change for a pendulum on the Moon (where g = 1.6 m/s²)?

Scientific explanations:

Christiaan Huygens (1629-1695), a Dutch scientist, developed the pendulum theory and invented the first accurate pendulum clock. The analysis of pendulum movement perfectly illustrates the principle of conservation of mechanical energy later formulated by Leibniz.

For a simple pendulum, this energy breaks down into two forms: gravitational potential energy, Ep = mgh (where h is the height relative to the lowest point), and kinetic energy, Ec = ½mv² (where v is the instantaneous speed). According to the conservation principle, the sum Ep + Ec remains constant in the absence of friction.

At the highest point of motion, the energy is mainly potential (v ≈ 0); at the lowest point it is mainly kinetic (h = 0). Between these extrema, energy gradually transforms from one form to another.

The FizziQ kinematic analysis module makes it possible to precisely quantify these transformations by providing the y coordinates (to calculate h) and the speed v at each instant. For an ideal frictionless pendulum, the total mechanical energy should remain perfectly constant.

In practice, there is a slight decrease over time, mainly due to air resistance and friction at the suspension point. This decrease is an example of energy dissipation, converted into heat according to the second law of thermodynamics.

The amplitude of the oscillations thus gradually decreases, a phenomenon called damping. The rate of this decrease can be used to estimate the coefficient of friction of the system.

Extension activities:

Frequently asked questions:

Q: My total energy varies a lot from point to point. Is conservation violated?
R: Large point-to-point variations are usually caused by noise in the velocity calculation, not by real energy changes. Velocity is computed from position differences, which amplifies pointing errors. Try using a larger interframe interval or smoothing the velocity data.

Q: Should I use the mass of the pendulum bob in the calculations?
R: If you only want to verify conservation (E_total is constant), you can set m = 1 and work in arbitrary energy units. The mass cancels when comparing Ep and Ec at different points. If you want absolute energy values, you need the actual mass.

Q: The pendulum in the video swings very fast. How do I track it accurately?
R: Use slow-motion playback if available. Track every frame when the bob is near the extremes (where it moves slowly and is sharp) but you may skip frames near the bottom (where it moves fastest and may be blurred). Use more frames overall for better accuracy.

Q: What amplitude should the pendulum have for the best results?
R: A moderate amplitude (15-30° from vertical) gives a good balance between having measurable height changes and staying in the small-angle regime where the motion is nearly sinusoidal. Very small amplitudes make height measurements difficult; very large amplitudes introduce non-linear effects.