Unveiling damped spring pendulum dynamics and constants through smartphone-integrated LiDAR sensors Open Access

While the practice of analyzing spring pendulums with smartphone accelerometers is well established,¹ the introduction of LiDAR sensor technology in smartphones opens new avenues of exploration.

LiDAR, an acronym for Light Detection And Ranging, is a technology used for measuring distances with high precision. It operates by emitting pulses of infrared laser light toward the target. Upon striking the object, the light pulses are reflected and captured by the LiDAR sensor. The sensor accurately determines the distance between itself and the target by calculating the time it takes for the light pulses to return. Unlike traditional measuring methods, LiDAR operates in real time and reliably and accurately measures distances, making it a powerful tool for various applications.

This description is among the first published smartphone experiments using the LiDAR sensor.² The accelerometer data usually collected in this experiment is somewhat abstract and may pose comprehension challenges for some students.³ In contrast, the LiDAR sensor, which measures distance instead of acceleration, might present a more intuitive alternative. This demonstrates the potential of smartphone-embedded LiDAR sensors for physics education, providing a novel approach to probing classical physics phenomena.

As the pendulum oscillates, the LiDAR sensor tracks the trajectory along the y-axis, vividly capturing the motion in this dimension. The exact distances measured are recorded and stored for later analysis.

The distance d at any given time t is described by Eq. (1).

We measured the mass of our smartphone, including the cable ties, and got m = 0.210 kg.

The collected data can be analyzed graphically or exported to a spreadsheet application like Microsoft Excel for thorough analysis. For the analysis, noise at the beginning and end of the experiment, which is caused by releasing and stopping the smartphone, should be removed. The different characteristics of the spring pendulum can be examined in two different ways. By interacting with the graph in the app itself, one can determine the period by selecting characteristic points such as adjacent maxima. By selecting a maximum and a minimum, the amplitude can be determined. Damping cannot be analyzed within the app. For more precise outcomes and to investigate damping effects, it is recommended to review the exported data for the identified local minima and maxima, which will yield more accurate values and facilitate the analysis of damping.

We aim to determine a spring pendulum’s spring constant and other relevant parameters using experimental data.

After exporting our data, we conducted an analysis using the Solver Add-In in Excel and the least-squares method. In addition to the column for the time and the measured deflections, we have added a column with the modeled values. These were based on the form of Eq. (1). In the next column of the table, we calculated the square of the difference between these two values. The initial values for A, λ, δ, and d₀ could be roughly estimated from the graphical representation in the app. With these first estimated values, we sum up the square of the individual differences from each measured value to the modeled value. This provides us with the least-squares value, an indicator of model fit. Our first least-squares value was over 80, indicating a bad fit. After using the Solver Add-In to optimize the fit, the least-squares value dropped to 0.11, therefore indicating a good fit.

The merger of smartphone technology and LiDAR has opened up a new, accessible method for investigating classical physics phenomena like the damped spring pendulum. Parallel to collecting data to analyze the damped spring pendulum with the LiDAR sensor, one could investigate the (damped) simple pendulum.