Center of gravity

This experience allows students to understand the concept of center of gravity and its behavior during a complex movement. It develops the capacity for precise observation and understanding of the fundamental laws of mechanics.

When a diver leaps from a 10-meter platform and performs a triple somersault, their body twists and rotates in seemingly chaotic fashion. Yet hidden within this spectacular motion is a point that traces a perfectly smooth parabolic arc through the air: the center of gravity. This concept, first rigorously defined by Archimedes in the 3rd century BC, is one of the most powerful ideas in mechanics. The center of gravity is the single point where the total gravitational force on a body can be considered to act. For a rigid object, it stays in a fixed position relative to the body. For an articulated body like a human, it shifts depending on posture and can even lie outside the body entirely, as when a high jumper arches over the bar using the Fosbury Flop technique. In diving, the center of gravity follows a clean parabola regardless of how the athlete tucks, pikes, or twists. This experiment uses FizziQ's kinematic analysis to track the center of gravity of a diver frame by frame and verify that it follows the predicted parabolic trajectory of a freely falling projectile.

Learning objectives:

The student uses the FizziQ Kinematic Analysis module to study the movement of a diver based on a video from the library. After calibrating the scale, he must identify and point on each image the position of the athlete's center of gravity, a point which can be located outside the body depending on the posture. By exporting this data to the experiment notebook, the student observes the shape of the trajectory and verifies that it corresponds to a parabola.

Level:

Middle and high school

FizziQ

Author:

Duration (minutes) :

40

What students will do :

- Identify and track the center of gravity of an articulated body (diver) during a complex movement
- Verify that the center of gravity follows a parabolic trajectory consistent with projectile motion
- Apply quadratic curve fitting to the trajectory data and extract physical parameters
- Understand the difference between the motion of the center of gravity and the motion of individual body parts
- Develop skills in video-based kinematic analysis and scale calibration

Scientific concepts:

- Center of gravity/center of mass, Parabolic motion, Gravity forces, Projectile trajectory, Barycenter

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, “Diving” video from the FizziQ library

Experimental procedure:

Open FizziQ and navigate to the Kinematics module. Load the 'Diving' video from the FizziQ resource library.
Identify a reference scale in the video. The height of the diving platform (typically 10 m, 5 m, or 3 m) or the height of the diver can be used. Set the scale in FizziQ.
Define the origin of the coordinate system at a convenient fixed point (e.g., the edge of the diving board or a corner of the frame).
Before starting the tracking, observe the video carefully. Note how the diver's body changes shape during the jump: the center of gravity shifts relative to the torso as limbs move.
For each frame, estimate the position of the center of gravity. For a standing position, it is roughly at the navel. When tucked, it shifts toward the center of the curl and may even lie outside the body.
Begin frame-by-frame pointing by clicking on the estimated center of gravity position in each successive frame. Track from the moment the diver leaves the board until just before entering the water.
Aim for at least 15-20 data points across the full trajectory.
After completing the tracking, view the y vs. x position graph in the FizziQ notebook. The trajectory should resemble a smooth, downward-opening parabola.
Use the FizziQ curve fitting tool to fit a quadratic function y = ax² + bx + c to the data.
From the coefficient a, estimate the acceleration of gravity: g = -2a × v₀ₓ², where v₀ₓ is the initial horizontal velocity.
Compare the smoothness of the center-of-gravity trajectory with the trajectory of a single body part (e.g., the feet or hands), which will show oscillations due to rotation.
Discuss the precision of your pointing and the sources of uncertainty in estimating the center of gravity position.

Expected results:

The center of gravity trajectory should form a smooth parabola with minimal scatter, even though the diver's body is rotating and changing shape. The quadratic fit should yield a coefficient of determination R² above 0.98 for careful tracking. The extracted value of g should be within 10-15% of 9.81 m/s², with discrepancies mainly due to the difficulty of precisely locating the center of gravity in each frame. By contrast, tracking a single body part (head, feet, or hands) will produce a trajectory with clear oscillations superimposed on the parabola, reflecting the rotational motion around the center of gravity. The horizontal velocity of the center of gravity should remain approximately constant throughout the flight, typically 1-3 m/s depending on the dive. Students may find that during tucked positions, the center of gravity can appear to lie outside the diver's body.

Scientific questions:

- Why does the center of gravity follow a smooth parabola while the diver's limbs follow much more complex paths?
- Can the center of gravity of a body lie outside the physical body? Give an example.
- How does the diver control the speed of rotation without any external force acting on them?
- Why is the Fosbury Flop technique in high jump advantageous from the perspective of center of gravity?
- What factors make it difficult to accurately locate the center of gravity in a video frame?
- How would air resistance affect the trajectory of the center of gravity during a dive?

Scientific explanations:

The center of gravity is the point of application of the resultant forces of gravity acting on a body. For an object subject only to gravity, this point follows a parabolic trajectory described by the equations of uniformly accelerated motion: x(t) = x₀ + v₀ₓt and y(t) = y₀ + v₀ᵧt - ½gt², where g is the acceleration of gravity (9.81 m/s²).

In the case of an articulated body like a diver, the center of gravity corresponds to the weighted barycenter of the masses of the different parts of the body. Its relative position is not fixed and can be located outside the physical body, depending on the posture adopted.

For example, when the diver curls or stretches, their center of gravity shifts relative to their anatomy, but the overall trajectory of that point remains invariably parabolic. The interest of this experience lies in the confrontation between visual intuition (complex rotational movement) and physical rigor (simple parabolic trajectory of the center of gravity).

Extension activities:

Frequently asked questions:

Q: How do I estimate the position of the center of gravity when the diver is in a tucked position?
R: When the diver is tucked, the center of gravity moves toward the geometric center of the curled body. It may even lie slightly outside the body, in the space enclosed by the arms and legs. Estimate the point equidistant from the major body masses (torso, thighs). Precision within a few centimeters is sufficient to see the parabolic pattern.

Q: My parabolic fit is not very good. What could be wrong?
R: The most common issue is inconsistent pointing of the center of gravity position. Try to use a systematic method for each frame. Also verify your scale calibration. If the diver rotates very fast, you may need to track every frame rather than skipping frames.

Q: The value of g I calculated is far from 9.81 m/s². Why?
R: The calculation of g from the quadratic coefficient is very sensitive to the accuracy of the scale calibration and the horizontal velocity estimate. A 10% error in the scale translates to a 20% error in g. Focus on verifying the parabolic shape of the trajectory rather than extracting a precise numerical value.

Q: Can I use this technique for other sports or activities?
R: Absolutely. Any activity involving airborne motion (gymnastics, long jump, trampoline, skateboard tricks) is suitable. The center of gravity always follows a parabola during free flight, regardless of how the body rotates or changes shape.