Lab report

A Sample Lab Report

Exp. 0: Empirical Equations

ABSTRACT

A study was done of the relationship between the diameters of various sized rings and their natural periods of oscillation when allowed to swing as a pendulum. Using the acquired data and the general form of the equation of a pendulum, , which is of the form: , values for the proportionality constant, A, and the power, n, were empirically determined. The numerical value for A was determined to be equal to 0.196, and a numerical value for n was determined to be 0.509. A comparison was done for the values of these same constants to that of a simple pendulum. A percent difference of 2.3% was determined for A and a percent difference of 1.8% was determined for n with this comparison. From this it is inferred that the equation for the period of a Ring Pendulum as a function of its diameter is of the same form as that for the period of a Simple Pendulum as a function of its length.

INTRODUCTION

Until a theoretical interpretation is worked out, the experimenter must use other methods to arrive at a systematic discussion of experimental data. The empirical method is based solely on experimental results. In this method, all variable factors are held constant except two. One of these is varied in a systematic fashion and corresponding values of the other factor are measured. In this way, one may soon conclude the fashion in which these variables are inter-dependent. The equation relating these variables is called an empirical equation.

In this experiment, the relation between the period of an oscillating ring and its diameter is to be determined. By utilizing rings of different diameters, one may arrive at the empirical equation relating the period and diameter for the rings.

From the basic knowledge of the behavior of a simple pendulum, a reasonable assumption to make might be that the period of a ring, T, is proportional to some power of the diameter, say n. Then it is seen that

(1)

Where A is the constant of proportionality. The experiment then will show whether this was a valid assumption.

Taking the logarithm of the above equation yields:

(2)

This is seen to be of the form:

(3)

Where the slope of the straight line will yield the parameter, n, and the y-intercept of the straight line will yield log A.

A plot of log T vs. log d using Excel is done to determine the slope and y-intercept of the trend line. In addition, a plot of T vs. d using a log-log scale is done as a comparison of determining n and A.

PROCEDURE

In this experiment, 5 rings, each having a different diameter, are used. Since each ring has a particular thickness, a mean value of each diameter is determined. Both the outer diameter and the inner diameter of each ring are measured in centimeters. For the two smaller rings, a Vernier caliper was used to measure the outer diameters and the inner diameters. For the three larger rings, a meter stick was used for these same measurements. Five measurements of the outside diameter do and five measurements of the inside diameter di for each ring are made with each measurement distributed uniformly around the ring to give an average value closer to the true value. The mean diameter d is the average of the outside and inside measurements.

Figure 1

RESULTS

The following results are taken from the log T vs. log d graph.

Comparison to simple pendulum:

	n	A
Simple Pendulum	0.5	0.2007089923
Ring Pendulum range values	[0.497…0.521]	0.196

The value for n for the Simple Pendulum falls within the range of n values for the Ring Pendulum. In addition, the value for A for the Ring Pendulum falls very nearly equal to the A value for the Simple Pendulum. Both of these experimentally found values are in agreement with the Simple Pendulum values.

Percent Error:

For n the percent difference between n for a Simple Pendulum and n empirically determined for the Ring Pendulum is:

% difference =

For A the percent difference between A for a Simple Pendulum and A empirically determined for the Ring Pendulum is:

% error =

It can be reasonably inferred from the data that the equation for a Ring Pendulum is of the same form as for the Simple Pendulum, substituting the diameter d for the length L in the equation.

Simple Pendulum:

Ring Pendulum:

Answers to Questions would appear here.

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