# Mathematical relationship

To investigate how the length factor of a pendulum string will affect the time period of a pendulum and to attempt and determine a mathematical relationship between the two. Hypothesis: A pendulum is made up of a mass (bob) attached to a string that is fastened so that the pendulum can swing (oscillate) in a plane. In a simple pendulum, all the mass is considered to be concentrated at a single point at the centre of the bob.

Physical quantities of the pendulum include length L, mass m, angle through which the pendulum swings , and the period T of the pendulum (which is the time it takes for the pendulum to swing through one complete oscillation). I am choosing to study the length L of the pendulum because past experience and preliminary investigation lead me to believe that the period of the pendulum should have a direct relationship with the length. From a Physics book, I have obtained the following theoretical expression for the period of a simple pendulum swinging in a plane: Where g is the acceleration due to gravity and the terms in the bracket are an infinite series.

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To find T for a given angle ? , the more terms you use, the greater the accuracy of the theoretical expression. That being said, for angles relatively small, it is sufficient enough to use the standard formula for the time period T, which is: Coming back to our research question, I predict that the length of a pendulum will certainly have an affect on the time period of a pendulum, and moreover, after investigating the formula above, I predict the relationship will be positive and exponential.In other words, if length is made four times greater, then the period should be only 2 times greater. I arrive at this hypothesis because if we look again at the relationship, and make 2? equal a certain constant k (and for us this is the case, since g, the acceleration due to gravity will remain constant), then we essentially have From this it becomes clear that the relationship between T and L is not linear but exponential and our results should show this.Variables: Independent – The independent variable in this experiment is the length of the string Dependant – we will be measuring the period of the pendulum, this is also the time taken to make one full swing. This measuring will be done by using a stop clock and will rely greatly on our vision and reaction time. We could not find an apparatus that would automatically record this dependant variable; however a stop clock is sufficient.

Control – The controls in this experiment are the mass of the pendulum and the angle at which it is dropped from, both of which will be kept constant. We will keep the mass constant simply by never changing the bob on the pendulum, and the we will keep the angle relatively controlled by measuring the same angle before every pendulum swing. The acceleration due to gravity, g, will be assumed to remain constant around 9. 81 ms-2 because we will perform the experiment and all trials in the same room.