1. Spring-Pendulum - Introduction
2.  
3. References

In this section we develop a mathematical model for a pendulum that connects to its pivot point with a spring. Many of the details on the physics of the spring and the pendulum are developed in the text by Haberman [1]. This section has a series of movies that show an actual filming of a spring pendulum with downloadable files for more details of a spring-pendulum. A discussion of this problem is presented for creating a mathematical model.

Spring Pendulum - Introduction

Below we present a series of movies for a spring-pendulum. The grid behind the pendulum measures 1.5 inches on the side of each square. There are 3 videos taken of the motion for this spring-pendulum. Click on the desired picture to see the video of the spring pendulum. (There will soon be a complete download of a frame-by-frame set of pictures for careful analysis of some inspired student or researcher.)

Simple Up-Down Motion
(Zip file with all still pictures and times for data analysis)

Side-to-side Motion
(Zip file with all still pictures and times for data analysis)

Complex Motion
(Zip file with all still pictures and times for data analysis)

The physical design of this spring pendulum uses a steel ball weighing 233 g attached to a brass spring (weighing 155 g). This configuration is mounted on a stand about 1 meter above the ground. The spring at rest has a total length of 18 cm (with the coiled part being 14 cm), while the ball is 3.5 cm in diameter. At rest with only the force of gravity acting on the ball, the length from the pivot point to the center of the steel ball is 53.5 cm. (Since the spring is slightly distended by its own weight, the steel ball stretches the spring 25.0 cm.)

A first experiment was performed with the steel ball being pulled down vertically. The motion was studied using a sonic motion detector that collected data 30 times a second. These data are available in an Excel file. Below is a graph showing the first 5 seconds of the data, shifted so that the motion is centered about the equilibrium.

A linear mathematical model for the vertical motion that is given in most Physics texts, where it is assumed that the spring is massless and all the mass is concentrated at a point, satisfies the damped spring equation

where y(t) represents the distance of the steel ball from the equilibrium position. The general solution of this equation is given by

where a = c/2m, w² = (4km - c²)/4m², A² = c₁² + c₂², and tan(z) = -c₂/c₁.

A nonlinear least squares best fit was performed on the complete data set in the Excel file noted above. (The best fit found that the data was shifted by 69.664 cm from the equilibrium, representing the distance of the recording device.) The other parameters found by this nonlinear least squares best fit were

It follows that A = 15.532 cm and z = -1.2210. The period of this pendulum is given by

References

[1] R. Haberman, Mathematical Models, Classics in Applied Mathematics, SIAM, 1998.