# Lab Report

**Topics:**Regression analysis, Measurement, Observational error

**Pages:**13 (2501 words)

**Published:**August 21, 2013

Department of Physics, University of Utah, Salt Lake City, 84112, UT, USA (Dated: March 6, 2013) Using a simple pendulum the acceleration due to gravity in Salt Lake City, Utah, USA was found to be (9.8 +/- .1) m/s2 . The model was constructed with the square of the period of oscillations in the small angle approximation being proportional to the length of the pendulum. The model was supported by the data using a linear ﬁt with chi-squared value: 0.77429 and an r-square value: 0.99988. This experimental value for gravity agrees well with and is within one standard deviation of the accepted value for this location.

I.

INTRODUCTION

The study of the motion of the simple pendulum provided valuable insights into the gravitational force acting on the students at the University of Utah. The experiment was of value since the gravitational force is one all people continuously experience and the collection and analysis of data proved to be a rewarding learning experience in error analysis. Furthermore, this experiment tested a mathematical model for the value of gravity that that makes use of the small-angle approximation and the proportional relationship between the square of the period of oscillations to the length of the pendulum.Sources of error for this procedure included precision in both length and time measurement tools, reaction time of the stopwatch holder, and the accuracy of the stopwatch with respect to the lab atomic clock. The ﬁnal result of g takes into account the correction for the error introduced using the approximation. There are opportunities to correct for the eﬀects of mass distribution, air buoyancy and damping, and string stretching[1]. Our results do not take these eﬀects into account at this time.

A.

Theoretical Introduction

The general form of Newton’s Law of Universal Gravitation can be used to ﬁnd the force between any two bodies. FG = −G mME ˆ 2 r RE (1)

2 On earth this equation can be simpliﬁed to F = −mgˆ with the value r GME 2 RE

taken to be the

constant g. The value of gravity in Salt Lake City (elev.1320 m) according to this model is: 9.81792 m/s2 [3][4][5]. The simple pendulum provides a way to repeatedly measure the value of g. The equation of motion from the free body diagram in Figure 1[2]:

FIG. 1: Free body diagram of simple pendulum motion[2].

F = ma = mgsinθ can be written in diﬀerential form ¨ g θ− θ=0 L The solution to this diﬀerential equation relies on the small angle approximation sinθ θ:

(2)

(3) θ for small

θ(t) = θ0 cos(

g ) L

(4)

3 The Taylor expansion

θ(t)

θo [1 −

gt2 g 2 t2 ] + 2L 4!L2

(5)

allows us to take the θ dependence out of the equation of motion. Taking the second derivative of the approximation gives the following:

g ¨ θ = −θ0 L

(6)

−θ0

g g g g + θ = 0 =⇒ θ0 = θ L L L L

g L,

(7)

4π 2 T2

˙ We know from the ﬁrst derivative θ = ω =

so it follows that since ω 2 =

=

g L

θ0 .

g 4π 2 =θ 2 L T

(8)

From the initial conditions it is also clear that the initial amplitude θ is equal to θ0 and so the linear relationship between length L and period T 2 can be expressed as

T2 = .

4π 2 L g

(9)

Using the small angle approximation introduces a small systematic error in the period of oscillation, T. For instance the maximum amplitude angle θ for a 1 percent error is .398 radians or 22.8 degrees; to reduce the error to 0.1 percent the angle must be reduced to .126 radians or 7.2 degrees. This experiment used an angle of about 10 degrees and that introduced an error of 0.3 percent. The calculations for the systematic error are found in the Appendix.

II.

EXPERIMENTAL PROCEDURE A. Setup

As seen in Figure 2, the pendulum apparatus was set up using a round metal bob with a hook attached to a string. The string passed through a hole in an aluminum bar, which...

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