Beginning with an entangled state of a time-independent (TI) quantum system coupled to its TI quantum environment, we show that a time-dependent Schrödinger equation (TDSE) for the quantum system alone can be derived in the limit that one of the environment variables becomes a classical variable. In the same limit the TI amplitude of the environment wavefunction becomes the TD amplitude of an eigenfunction expansion of the system TD wavefunction. Similarly, the phase of the TI environment wavefunction goes over into the TD phase of the system amplitude. By considering that more and more environment variables become successively classical, each provides a classical clock to give a multiple-time TDSE for the quantum system. Two examples of two-time TDSE given in the literature are derived. When all clocks are synchronized the multiple-time TDSE reduces to the usual one-time TDSE.

This paper describes the use of smartphone’s gyroscope sensor to analyse a hollow cylinder rolling down an inclined plane. The smartphone (iPhone 4s) was attached to the end of hollow cylinder and was equipped with the Sensorlog application (Sensorlog app) to record the angular speed of rolling down an inclined plane. The experimental results agree with the theoretical model that is familiar to students for the rolling motion on an inclined plane. Moreover, the coefficients of static friction and kinetic friction were determined to be 0.205 ± 0.011 and 0.178 ± 0.003 from the measurements, respectively. This experiment demonstrated an alternative way to teach the rolling motion in a physics laboratory.

This study used a smartphone's digital compass to observe the rolling motion of a hollow cylinder on an inclined plane. The smartphone (an iPhone 4s) was attached to the end of one side of a hollow cylinder to record the experimental data using the SensorLog application. In the experiment, the change of angular position was measured by the smartphone's digital compass. The obtained results were then analyzed and calculated to determine various parameters of the motion, such as the angular velocity, angular acceleration, critical angle, and coefficient of static friction. The experimental results obtained from using the digital compass were compared with those obtained from using a gyroscope sensor. Moreover, the results obtained from both sensors were consistent with the calculations for the rolling motion. We expect that this experiment will be valuable for use in physics laboratories.

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This paper demonstrates a physics laboratory experiment for determining the coefficient of static friction from circular motion using a smartphone's sensors. The coefficient of static friction between the surface of a smartphone and a turntable can be determined from the limiting value of the static friction that keeps the smartphone in a circular path. The smartphone was used to record an angular speed with an appropriate application (app) to determine the limiting value of angular speed in a circular motion. This limiting value was further calculated to obtain the static friction coefficient. We expect this experiment to be useful for physics teachers to help demonstrate the topic of circular motion.