The Simple Pendulum Laboratory Report

This is the Simple Pendulum Laboratory Report.

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Abstract

The pendulum method is used for determination of the acceleration of gravity (g). The thin string used and a large mass reduces frictional effects and air drag. The long pendulum arm and a small swing about a small angle help in the approximation of the simple harmonic motion. The slope of the graph was 0.23 while a graphical value of the acceleration of gravity was g=9.0824. The acceleration of gravity from the calculation was g = 9.7744.

Hypothesis

As the length of the string increases, the period (T) of oscillation increases

Introduction

In pendulum method, the period of oscillations is independent of the pendulum mass, but dependent of the square root of the string length. The simple pendulum setup can be used for the determination of acceleration of gravity value (g) (Cutnell, & Kenneth, 2013). The mass of the pendulum should be kept constant while the length of the string is varied. The length is the manipulated variable, period (T) a responding variable while the mass of the pendulum a fixed variable.

L=lengt

M= mass

X=amplitude

g = acceleration of gravity

1. From Newton’s 2^nd law of motion

F= dp/dt

1. Hooke’s law

F = -kx, k is the spring constant and x is the displacement

Therefore, the back and forth motion can be expressed as

w=2Πf =

Replacing the mass with a moment of inertia (I), then the equation becomes;

w=2Πf = )

mw²=k

For small angles of displacement,

The equation below gives the time period in terms of the angular frequency w

Substituting for w, then

The acceleration of gravity should be small to ascertain the stability of the system during oscillation and also enable accurate determination of the period. A small angle of oscillation should be used so that the system exhibits simple harmonic motion (Cutnell, & Kenneth, 2013).

Experimental Set-Up

The clamp was attached to the table in a stable position. The mass was attached to a string, and set to settle with respect to the pivot point before starting oscillations. The mass was made to hang freely at zero degree in respect to the clamp. The smart timer was set on the pendulum. The oscillations of the mass were started, and the period of a complete oscillation recorded. The pendulum was set to oscillate with a small angle. The length of the string was increased in steps and the period for every oscillation recorded.

Experimental Analysis

Table: 1. Table showing changes in period (T) as time increases

Figure: 1. Graph of Length versus Square Period (T²)

Statistical Analysis

The simple pendulum equation is given as;

T = 2Π ( )

From the graph

Slope =   =  = 0.23

Therefore, g = 4Π² × 0.23 = 9.0824

From calculations

Let L = 1 m and T = 2.020 seconds

T = 2Π ( )

Therefore, g =   =   = 9.7744

Percentage difference  × 100 = 0.070 %

Therefore, the slope coefficient from the graph was 0.23; the statistical acceleration of gravity was 9.0824. On the other hand, the calculated value for the acceleration of gravity was 9.7744 indicating a difference of approximately 0.070%.

Conclusion

A simple pendulum was set up consisting of a mass (m) and a negligible string of length (L) attached on a frictionless pivot. The mass was made to oscillate and period recorded on smart timer. The results obtained were used for plotting a graph of L against T² to determine the slope coefficient. The w = 2Πf = ( ) equation was used to determine the acceleration of gravity (g). The pendulum method is the best for the determination of acceleration of gravity (g) as compared to the ball drop method because there is no transformation of energy (gravitational potential energy). The simple pendulum method leads to an approximately accurate value of the acceleration of gravity (g) as compared to the ball drop method. The experiment was successful since the acceleration of gravity value obtained was close to the accepted value of 9.806. The sources of error were estimations while reading of value meter values, calculation and plotting of the graph. The hypothesis was correct.

Acknowledgements

I acknowledge the physics department and laboratory staff for availing the laboratory facilities.

I acknowledge the laboratory staff for providing directions on how to carry out the experiment.

I acknowledge our tutor for teaching and guidance through the course.

I acknowledge my fellow students for the support and discussions concerning the topic and the experiment.

References

Cutnell, D. J., & Kenneth, W. J. (2013). Introduction to Physics. London: John Wiley &    Sons, Limited.

APPENDIX

Crude Results