^{1}, Danielle Holz

^{1}, Alae Kawam

^{1}and Mary Lamont

^{1}

### Abstract

The advent of new sensor technologies can provide new ways of exploring fundamental physics. In this paper, we show how a Wiimote, which is a handheld remote controller for the Nintendo Wii video game system with an accelerometer, can be used to study the dynamics of circular motion with a very simple setup such as an old record player or a bicycle wheel.

We would like to thank Stephen Takacs for the help with the experimental setups; James Supplee, Robert Murawski, and Adam Fanslau for helpful discussions; and various open source communities.

Experiment 1: Vector addition

Experiment 2: Magnitude of the acceleration

Experiment 3: Vertical circular motion

Experiment 4: Non-uniform circular motion

Discussion

### Key Topics

- Kinematics
- 15.0
- Acceleration measurement
- 6.0
- Friction
- 5.0
- Computer software
- 2.0
- Data analysis
- 2.0

## Figures

Mechanical analogy of the Wiimote operation for three different cases. (a) When there is no gravitational force or acceleration, the mass inside the box is sitting in the center. The spring force is measured to be zero. (b) When the box is at rest on the ground, the internal mass will be shifted from the center due to gravity, and there is an upward spring force, , denoted by the arrow. (c) When the box is rotating horizontally (i.e., gravity is not involved), the spring force provides the centripetal acceleration toward the center of the circular motion. Hence, the internal force is directly related to the gravitational force or the acceleration of the entire box.

Mechanical analogy of the Wiimote operation for three different cases. (a) When there is no gravitational force or acceleration, the mass inside the box is sitting in the center. The spring force is measured to be zero. (b) When the box is at rest on the ground, the internal mass will be shifted from the center due to gravity, and there is an upward spring force, , denoted by the arrow. (c) When the box is rotating horizontally (i.e., gravity is not involved), the spring force provides the centripetal acceleration toward the center of the circular motion. Hence, the internal force is directly related to the gravitational force or the acceleration of the entire box.

A bird's-eye view of the setup for the uniform circular motion experiment. The vector addition diagram was drawn according to the Wiimote measurements, which were (−6.3, 9.9, −3.0) and (−0.6, 10.2, −6.3) m/s^{2} for the left and right plots, respectively. The starting point of each vector is set to the left of the A button, where the accelerometer chip is located. The resultant vector always points along the radial direction, as expected.

A bird's-eye view of the setup for the uniform circular motion experiment. The vector addition diagram was drawn according to the Wiimote measurements, which were (−6.3, 9.9, −3.0) and (−0.6, 10.2, −6.3) m/s^{2} for the left and right plots, respectively. The starting point of each vector is set to the left of the A button, where the accelerometer chip is located. The resultant vector always points along the radial direction, as expected.

Uniform circular motion experiment. The error bars (smaller than the markers) indicate the standard deviation of three trials. The regression line for the data is shown in black, and the expected result is in red.

Uniform circular motion experiment. The error bars (smaller than the markers) indicate the standard deviation of three trials. The regression line for the data is shown in black, and the expected result is in red.

Ferris wheel setup. The Wiimote (green rectangle) rotates in the counterclockwise direction and measures the accelerations due to gravity, radial centripetal force, and tangential frictional force.

Ferris wheel setup. The Wiimote (green rectangle) rotates in the counterclockwise direction and measures the accelerations due to gravity, radial centripetal force, and tangential frictional force.

Ferris wheel experiment. In the top plot, the radial (red) and tangential (blue) accelerations are graphed as a function of time. The acceleration perpendicular to the plane of rotation (yellow) is zero. The period of each revolution *T*is determined as the time between the successive minima in the red curve (⋆ symbols). The tangential acceleration due to friction is present (as evident in the increasing periods) but relatively small. The average angular speed , and the average radial acceleration <*a*> is calculated during the corresponding period. The instantaneous radial acceleration is equal to , so and close to , as shown by the bottom two plots. The red lines show the expected results with , and the black regression lines are the fits to the data.

Ferris wheel experiment. In the top plot, the radial (red) and tangential (blue) accelerations are graphed as a function of time. The acceleration perpendicular to the plane of rotation (yellow) is zero. The period of each revolution *T*is determined as the time between the successive minima in the red curve (⋆ symbols). The tangential acceleration due to friction is present (as evident in the increasing periods) but relatively small. The average angular speed , and the average radial acceleration <*a*> is calculated during the corresponding period. The instantaneous radial acceleration is equal to , so and close to , as shown by the bottom two plots. The red lines show the expected results with , and the black regression lines are the fits to the data.

Deceleration due to friction during horizontal rotation. The radial and tangential accelerations are plotted on the left, and from these measurements the angular velocity ω and *d*ω/*dt*, respectively, are calculated. As shown on the right plot, the frictional torque is proportional to the second power of ω plus some constant. Three trials are shown in different colors, and each data point is a 3-s average of the Wiimote measurements.

Deceleration due to friction during horizontal rotation. The radial and tangential accelerations are plotted on the left, and from these measurements the angular velocity ω and *d*ω/*dt*, respectively, are calculated. As shown on the right plot, the frictional torque is proportional to the second power of ω plus some constant. Three trials are shown in different colors, and each data point is a 3-s average of the Wiimote measurements.

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