Applying the Model to the Graph

Oct 13, 2025 by wbrum.org 148 views

**Analyzing the Motion of a Bicycle Wheel: A Wireless Sensor's Perspective**

Understanding the Graph

The graph provided shows the sensor's height above the ground, in centimeters, over time t, measured in seconds. To analyze this motion, we need to understand the key features of the graph.

Key Features of the Graph

Periodic Motion: The graph shows a periodic motion, indicating that the bicycle wheel is spinning in a circular motion.
Amplitude: The amplitude of the graph represents the maximum height of the sensor above the ground, which is approximately 30 centimeters.
Frequency: The frequency of the graph represents the number of cycles or rotations of the wheel per second, which is approximately 2 Hz.
Phase Shift: The phase shift of the graph represents the initial position of the wheel, which is not at the top or bottom of the rotation.

Mathematical Analysis

To analyze the motion of the bicycle wheel, we can use mathematical models. One such model is the circular motion model, which describes the position of an object moving in a circular path.

Circular Motion Model

The circular motion model is described by the following equations:

x(t) = r cos(ωt + φ): This equation describes the horizontal position of the object as a function of time.
y(t) = r sin(ωt + φ): This equation describes the vertical position of the object as a function of time.

where:

r is the radius of the circular path
ω is the angular frequency (related to the frequency of the motion)
φ is the phase shift (initial position of the wheel)

Applying the Model to the Graph

To apply the circular motion model to the graph, we need to determine the values of r, ω, and φ.

r: The radius of the circular path is equal to the amplitude of the graph, which is approximately 30 centimeters.
ω: The angular frequency is related to the frequency of the motion, which is approximately 2 Hz. We can calculate ω using the following equation: ω = 2πf, where f is the frequency.
φ: The phase shift is not explicitly given in the graph, but we can estimate it by looking at the initial position of the wheel.

Calculating the Values

Using the equations above, we can calculate the values of r, ω, and φ.

r = 30 cm
ω = 2π(2 Hz) = 12.57 rad/s
φ = 0 (assuming the initial position of the wheel is at the top of the rotation)

Conclusion

In conclusion, the graph provided shows a periodic motion of a bicycle wheel spinning freely on its axle. By analyzing the key features of the graph and applying the circular motion model, we can determine the values of r, ω, and φ. This analysis provides a deeper understanding of the motion of the bicycle wheel and can be used to make predictions about its behavior.

Q&A

Q: What is the amplitude of the graph? A: The amplitude of the graph is approximately 30 centimeters.

Q: What is the frequency of the graph? A: The frequency of the graph is approximately 2 Hz.

Q: What is the phase shift of the graph? A: The phase shift of the graph is not explicitly given, but we can estimate it by looking at the initial position of the wheel.

Q: How can we calculate the values of r, ω, and φ? A: We can calculate the values of r, ω, and φ using the equations of the circular motion model.

Q: What is the significance of the circular motion model in this context? A: The circular motion model provides a mathematical framework for understanding the motion of the bicycle wheel and making predictions about its behavior.

Q: Can we use this analysis to make predictions about the motion of the bicycle wheel? A: Yes, we can use this analysis to make predictions about the motion of the bicycle wheel. By understanding the key features of the graph and applying the circular motion model, we can make predictions about the behavior of the wheel in different scenarios.