Kater’s Pendulum
Kater’s Pendulum
Thomas Markovich and Kapil Chhabria
Departments of Physics University of Houston Houston, TX 77204-5006 (Dated: December 9, 2010)
We experimentally determined the local gravitational constant using Kater’s Pendulum to provide the authors with experience in data analysis. In this manuscript, we rigorously derive the relevant equations from first principles with the appropriate expressions for the experimental uncertainty. We found that by assuming the periods were equal we were able to determine that g was 9.7993 ± 0.0010 m/s2 which was 0.0653 % from the known value. If we took into account the period difference, we found that g was 9.7982 ± 0.0083 m/s2 which is 0.0552 %. Both methods gave us results well within our goal of 0.2% deviation from the expected value.
I.
INTRODUCTION
is equal to τ = I0 α (II.2)
In the course of the author’s physics degree, no constant has been so ubiquitous as the local acceleration due to gravity. As such, it proves an important pedagogical exercise to experimentally verify its value. To do this, we will repeat the Kater’s pendulum experiment[1]. Our goal in this experiment is to determine the local acceleration due to gravity to within 0.2% of the accepted value of 9.7928m/s2 [1]. The remainder of this paper will be organized as follows. In Section II, we will provide a brief introduction to the theory with rigorous derivations of the propagation of error equations. In Section III, we will provide an outline of our experimental procedure with diagrams as necessary to aid in clarity. Section IV will contain our reported results, error analysis, and discussion of any possible conflicts with theory. Finally, in Section V, we will provide the reader with a summary of our experiment with major conclusions.
where I0 is the moment of inertia about the point of oscillation and α is simply the angular acceleration. We note that this ultimately yields the following differential equation −M dg sin θ d2 θ = dt2 I0
Thomas Markovich and Kapil Chhabria
Departments of Physics University of Houston Houston, TX 77204-5006 (Dated: December 9, 2010)
We experimentally determined the local gravitational constant using Kater’s Pendulum to provide the authors with experience in data analysis. In this manuscript, we rigorously derive the relevant equations from first principles with the appropriate expressions for the experimental uncertainty. We found that by assuming the periods were equal we were able to determine that g was 9.7993 ± 0.0010 m/s2 which was 0.0653 % from the known value. If we took into account the period difference, we found that g was 9.7982 ± 0.0083 m/s2 which is 0.0552 %. Both methods gave us results well within our goal of 0.2% deviation from the expected value.
I.
INTRODUCTION
is equal to τ = I0 α (II.2)
In the course of the author’s physics degree, no constant has been so ubiquitous as the local acceleration due to gravity. As such, it proves an important pedagogical exercise to experimentally verify its value. To do this, we will repeat the Kater’s pendulum experiment[1]. Our goal in this experiment is to determine the local acceleration due to gravity to within 0.2% of the accepted value of 9.7928m/s2 [1]. The remainder of this paper will be organized as follows. In Section II, we will provide a brief introduction to the theory with rigorous derivations of the propagation of error equations. In Section III, we will provide an outline of our experimental procedure with diagrams as necessary to aid in clarity. Section IV will contain our reported results, error analysis, and discussion of any possible conflicts with theory. Finally, in Section V, we will provide the reader with a summary of our experiment with major conclusions.
where I0 is the moment of inertia about the point of oscillation and α is simply the angular acceleration. We note that this ultimately yields the following differential equation −M dg sin θ d2 θ = dt2 I0