Cycloid

This activity allows students to discover the mathematical curve that is the cycloid and its geometric properties. It develops the ability to analyze complex movement using digital tools.

In 1696, Johann Bernoulli posed a challenge to the mathematicians of Europe: what is the shape of the curve along which a ball rolls from one point to another in the shortest possible time? The answer, discovered independently by Newton, Leibniz, and others, was the cycloid, the curve traced by a point on the rim of a rolling wheel. This elegant mathematical curve had already fascinated Galileo, who called it the most graceful of curves, and Pascal, who reportedly solved cycloid problems during a bout of insomnia to distract himself from a toothache. The cycloid possesses remarkable properties: it is both the brachistochrone (shortest time path under gravity) and the tautochrone (a ball released from any point on the curve reaches the bottom in the same time). These properties made it central to the development of calculus and mechanics. In this experiment, students use FizziQ's kinematic analysis to track a point on the rim of a moving bicycle wheel and discover the cycloid firsthand, connecting abstract mathematical equations to a visible, physical motion.

Learning objectives:

The student studies the trajectory of a point located on the edge of a moving bicycle wheel using the FizziQ kinematic analysis module. From a real video or from the application library, the student performs precise pointing of the movement frame by frame then analyzes the shape of the curve obtained, its periodicity characteristics and its maximum height relative to the ground.

Level:

High school

FizziQ

Author:

Duration (minutes) :

40

What students will do :

- Track the trajectory of a point on the rim of a rolling wheel using video-based kinematic analysis
- Identify the resulting curve as a cycloid and describe its geometric properties
- Measure the period and amplitude of the cycloid and relate them to the wheel radius
- Understand the composition of translational and rotational motion that generates the cycloid
- Compare the experimental curve with the theoretical parametric equations x = r(t - sin t), y = r(1 - cos t)

Scientific concepts:

- Cycloid
- Rotational and translational movement
- Periodicity
- Curve configuration
- Kinematics of Compound Motion

Sensors:

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

What is required:

- Smartphone with the FizziQ application
- Video of a moving bicycle or use of the 'Cycloid' video from the FizziQ library
- Experience notebook for analysis of results

Experimental procedure:

Open FizziQ and navigate to the Kinematics module. Load the 'Cycloid' video from the FizziQ resource library, or use a video of a bicycle wheel with a visible marker on the rim.
Identify a reference scale in the video. The wheel diameter is ideal if known. Set the scale in FizziQ.
Set the origin of the coordinate system at the initial position of the marked point on the rim when it touches the ground.
Begin frame-by-frame tracking by clicking on the marked point on the rim in each successive frame.
Track through at least two complete revolutions of the wheel (the point should touch the ground twice after the starting position).
Aim for at least 20-30 data points per revolution for a smooth curve.
After completing the tracking, view the y vs. x position graph in the FizziQ notebook. You should see the characteristic arch-shaped cycloid curve.
Measure the horizontal distance between two consecutive cusps (points where the curve touches the x-axis). This distance should equal 2πr, the circumference of the wheel.
Measure the maximum height of the curve. It should equal 2r, twice the wheel radius.
Verify the relationship: horizontal period = π × maximum height (since 2πr = π × 2r).
Use the FizziQ tools to compare your experimental curve with the theoretical cycloid equations: x = r(θ - sin θ) and y = r(1 - cos θ).
Note the cusp points where the rim point touches the ground: the velocity is momentarily zero at these points, which is visible as a clustering of data points.

Expected results:

The position-position plot should show the characteristic arch shape of a cycloid, with smooth arches meeting the x-axis at sharp cusps. The horizontal distance between consecutive cusps should equal the wheel circumference (2πr), and the maximum height should equal the wheel diameter (2r). For a typical bicycle wheel with radius 33 cm, the arch height should be about 66 cm and the horizontal period about 207 cm. Data points will cluster near the cusps (where the point moves slowly) and be more widely spaced at the top of the arch (where the point moves fastest, at twice the wheel's translational speed). The experimental curve should match the theoretical cycloid well, with deviations mainly due to pointing precision and slight wheel wobble.

Scientific questions:

- Why does the marked point momentarily stop at each cusp of the cycloid?
- What is the speed of the marked point at the top of the arch relative to the ground? How does it compare to the speed of the wheel's axle?
- Why is the cycloid called the brachistochrone curve? What does this mean physically?
- How would the curve change if the marked point were located halfway between the rim and the axle instead of on the rim?
- What is the total length of one arch of a cycloid? How does it compare to the circumference of the wheel?
- How did the cycloid contribute to the development of calculus in the 17th century?

Scientific explanations:

The cycloid is a mathematical curve described by a point fixed on the circumference of a circle which rolls without sliding on a horizontal line. Parametrically, for a circle of radius r, it is expressed by: x(t) = r(t - sin t) and y(t) = r(1 - cos t).

This curve has several remarkable properties: its maximum height is equal to 2r (twice the radius of the circle) and its horizontal period is equal to 2πr. Historically, it has fascinated mathematicians like Pascal and Huygens for its properties of brachistochrone (fastest descent curve) and tautochrone (descent time independent of the starting point).

The FizziQ kinematic analysis tool makes it possible to experimentally verify these theoretical properties by following the real trajectory of a point on a wheel point by point. When the tracking point is moved towards the inside of the wheel (on a spoke), the resulting curve becomes a shortened cycloid (or trochoid).

At the center of the wheel, the movement becomes a simple uniform rectilinear translation. This experiment perfectly illustrates the composition of a rotation and translation movement, a fundamental concept in kinematics.

Extension activities:

Frequently asked questions:

Q: My tracked curve does not show sharp cusps at the bottom. What might be wrong?
R: The cusps occur when the marked point touches the ground, and the velocity there is very small, so many frames cluster at this location. If you are skipping too many frames, you may miss the cusp region. Try tracking every frame near the bottom of the trajectory.

Q: The maximum height of my curve does not equal twice the wheel radius. Why?
R: Check your scale calibration carefully. If the camera is not perfectly perpendicular to the plane of the wheel, perspective distortion can alter the apparent dimensions. Also verify that you are tracking a point truly on the outer rim, not on a spoke or hub.

Q: What is the difference between a cycloid and a sine wave? They look similar.
R: While both are periodic and arch-shaped, the cycloid has sharp cusps where it meets the baseline, whereas a sine wave has smooth, rounded troughs. The cycloid arch is also taller relative to its width than a sine arch of the same period.

Q: Can I observe a cycloid in everyday life without a bicycle?
R: Cycloids appear in many contexts: the trajectory of a point on a rolling ball, the design of gear teeth (involute curves are related), and even in the splash patterns of water drops on a moving surface.