1-1. Simple Pendulum Motion Simulation
1-2. Analysis Results of Simple Pendulum Motion Simulation
The results that can be obtained by analyzing a simple pendulum motion experiment are as follows:
- The period of a simple pendulum motion does not change with the mass’s size.
- The period of a simple pendulum motion increases slightly as the amplitude of the pendulum increases, but the difference is not significant.
- In simple pendulum motion, the centripetal force does not act at the start, but it is maximal at the point where the speed is the fastest—at the center of oscillation. Therefore, the probability of the string breaking is highest at the center of oscillation, where the tension is greatest, and the string does not break at the maximum height.
- The force used in the simple pendulum motion equation is not the net force but the tangential component of the net force.
2-1. Mathematical Explanation of Pendulum Motion
The differential equation
[ \theta''(t) + \frac{g}{l} \sin(\theta(t)) = 0 ]
describes the angle ( \theta(t) ) of a pendulum at time ( t ), where ( g ) is the acceleration due to gravity and ( l ) is the length of the pendulum, with ( \theta”(t) ) being the angular acceleration. This equation explains how the pendulum swings at some angle relative to the vertical. As the pendulum moves, it experiences a restorative force due to gravity, which is proportional to the angle of displacement from the vertical line. Hence, the pendulum tends to return to its original position.
However, since this equation is nonlinear, it is not possible to find a general solution. When analyzing the motion of a pendulum at small angles, the ( \sin(\theta(t)) ) can be approximated to ( \theta(t) ), allowing for linearization. This is known as the ‘small-angle approximation,’ and in this case, the motion equation simplifies to:
[ \theta''(t) + \frac{g}{l} \theta(t) = 0 ]The concept of simple pendulum motion that is taught in high school often simplifies explanations to aid students’ understanding, which can lead to misconceptions. For those who decide to major in physics, revisiting mechanical phenomena through mathematical physics is a good way to deepen their understanding.