In orbit

This activity allows students to experimentally verify the relationship between centripetal acceleration and angular velocity. It makes the link between the theoretical equations of circular motion and real measurements.

Every satellite orbiting Earth, from the International Space Station to GPS satellites, stays in orbit because of one fundamental force: centripetal acceleration. This acceleration, directed toward the center of the circular path, keeps the object curving instead of flying off in a straight line. Isaac Newton realized that the same force that pulls an apple to the ground also holds the Moon in orbit, differing only in magnitude. The relationship between centripetal acceleration, speed, and radius is elegantly simple: a = v²/r = ω²r. Yet this equation has profound implications, governing everything from the spin cycle of a washing machine to the design of highway curves. In this experiment, students become their own centrifuge by holding a smartphone at arm's length and spinning around. The phone's accelerometer measures the centripetal acceleration directly, and by comparing this with the independently measured angular velocity and radius, students can verify the fundamental equation of circular motion with their own bodies.

Learning objectives:

The student holds his smartphone at arm's length in a vertical position and turns around while recording the normal acceleration with the FizziQ accelerometer. After having made a few complete rotations, the student analyzes the acceleration graph to calculate the average centripetal acceleration and the angular velocity then experimentally verifies the relationship a = ω²r.

Level:

High school

FizziQ

Author:

Duration (minutes) :

25

What students will do :

- Measure centripetal acceleration during circular motion using the smartphone accelerometer
- Independently determine the angular velocity by timing the rotations
- Verify the relationship a = ω²r between centripetal acceleration, angular velocity, and radius
- Understand the direction and magnitude of centripetal acceleration in uniform circular motion
- Connect the experimental results to orbital mechanics and real-world applications

Scientific concepts:

- Uniform circular movement
- Centripetal acceleration
- Angular velocity
- Frenet’s landmark
- Relationship between vector quantities

Sensors:

- Accelerometer (normal acceleration or total acceleration)

What is required:

- Smartphone with the FizziQ application
- Clear space to turn safely
- Tape measure to measure arm length
- FizziQ experience notebook

Experimental procedure:

Measure the radius of rotation: with your arm fully extended holding the smartphone, measure the distance from the center of your body to the phone. Record this as r (typically 0.6-0.9 m).
Find a clear, open space where you can safely turn around with arms extended without hitting anything.
Open FizziQ and select the Accelerometer sensor. Choose the component perpendicular to the phone screen (often labeled normal or Y-axis), which will measure the centripetal acceleration.
Hold the phone vertically at arm's length with the screen facing you. The centripetal acceleration will be directed along the axis pointing from the phone toward your body.
Start recording in FizziQ. Begin spinning slowly at a steady rate. Complete at least 5 full rotations at this speed.
Have a partner count your rotations and time them with a stopwatch, or count them yourself. Record the time for 5 rotations.
Stop spinning and stop recording. Note the average acceleration during the steady spinning phase from the graph.
Calculate the angular velocity: ω = 2π × (number of rotations) / (total time in seconds).
Calculate the theoretical centripetal acceleration: a_theory = ω² × r.
Compare the measured and theoretical values. They should agree within 10-20%.
Repeat at a faster rotation speed and verify that the acceleration increases with ω².
Repeat at different arm lengths (bent arm vs. fully extended) to investigate the effect of radius on centripetal acceleration at the same angular velocity.

Expected results:

For a typical arm length of 0.75 m and a comfortable rotation rate of about 1 revolution per second (ω ≈ 6.28 rad/s), the expected centripetal acceleration is approximately 30 m/s² (about 3g). Faster spinning (1.5 rev/s) would produce about 67 m/s² (7g). The measured acceleration should agree with the theoretical value within 10-20%, with discrepancies arising from non-uniform rotation speed, difficulty in maintaining a perfectly circular path, and the phone not being exactly perpendicular to the radius. Students should clearly observe that doubling the angular velocity quadruples the acceleration, confirming the ω² dependence. The effect of changing the radius should also be visible: a shorter arm (0.4 m) at the same angular velocity produces proportionally less acceleration.

Scientific questions:

- Why must an object moving in a circle experience an acceleration even if its speed is constant?
- In what direction does the centripetal acceleration point? What provides the centripetal force in this experiment?
- How does the centripetal acceleration of a satellite in orbit relate to the gravitational force?
- What would happen if the centripetal force suddenly disappeared while you were spinning?
- Why does a higher rotation speed produce a much larger acceleration (quadratic relationship)?
- How is centripetal acceleration used in the design of centrifuges and amusement park rides?

Scientific explanations:

In uniform circular motion, any point follows a circular path at constant speed. Although the speed is constant in norm, its direction changes continuously, which implies an acceleration perpendicular to the trajectory, directed towards the center of the circle: this is centripetal acceleration.

This acceleration is linked to the tangential speed v and the radius r by the formula a = v²/r. In terms of angular velocity ω (in rad/s), this relationship becomes a = ω²r, where ω = 2πf with f the rotation frequency in Hz.

The smartphone's accelerometer measures this acceleration when it is held perpendicular to the plane of rotation (in the Frenet frame, normal acceleration corresponds to centripetal acceleration). For this experiment, the radius corresponds to the length of the outstretched arm (typically 60-70 cm).

The angular velocity can be determined by counting the number of revolutions made during the duration of the recording, or by analyzing the periodicity of the acceleration signal. Sources of error include: radius variation during movement, inaccuracy in smartphone orientation, and irregularity in rotation speed.

This experiment illustrates a fundamental principle of celestial mechanics: it is this same centripetal acceleration that keeps the planets in orbit around the Sun, although in this case it is produced by gravitational force.

Extension activities:

Frequently asked questions:

Q: The acceleration reading is very noisy and keeps changing. How do I get a stable measurement?
R: Maintaining a perfectly constant rotation speed is difficult for a human. Spin at a comfortable, sustainable pace and look for the average value during the steadiest portion of the recording. Having a partner count rotations helps maintain consistent speed.

Q: Which axis of the accelerometer measures the centripetal acceleration?
R: It depends on how you hold the phone. If the screen faces you while spinning, the centripetal acceleration is along the axis perpendicular to the screen (often labeled Z or normal). Try different axes to find the one that shows the largest and most consistent reading during rotation.

Q: Why do I feel dizzy after spinning? Is this related to centripetal acceleration?
R: Dizziness is caused by the fluid in your inner ear continuing to rotate after you stop, confusing your balance system. The centripetal acceleration itself is not directly responsible, but the rotation that produces it is.

Q: Can I measure centripetal acceleration in a car going around a curve?
R: Yes. The phone's accelerometer will register a lateral acceleration during turning. This is the centripetal acceleration provided by the friction between the tires and the road. Typical values are 2-5 m/s² for normal driving turns.