The math behind the Pendulum - Research Paper Example

As the period of a pendulum is constant, pendulums were used to regulate the movement of clocks. Until the 1930’s pendulums were the most accurate time keeping devices of the world. In 1583, the Italian scientist Galileo first noted the constancy of a pendulum’s period by comparing the movement of a swinging chandelier in a Pisa cathedral with his pulse rate. He found that the time was not a function of how wide the chandelier swung. As the wind was blowing the chandelier, it was swinging different distances side to side or amplitudes.

Galileo found that the pendulum swung more slowly. Over a shorter swing, the chandelier took just as many of his heartbeats to complete a swing with greater amplitude. Galileo made an error in the calculation of the angle of the chandelier. In 1656, the Dutch mathematician and scientist Christian Huygens invented a clock controlled by the motion of a pendulum (Huygens and the Pendulum, Princeton). The accuracy of mechanical clocks improved in the span of a couple of decades in the early 17th century from plus or minus half an hour per day to one second per day.

This quantum increase in accuracy of timing enabled previously unimagined degrees of precision measurement in mechanics, astronomy and other fields of study. Time then for the first time was expressed as an independent variable in the investigation of nature. For example, each of the following could be reliably investigated for the first time: The effect of force on objects over time The distance of fall over time The change of speed over time The radial movements of planets over time The progress of chemical reactions over time All these investigations required that the time could be accurately and reliably measured.

Thus the pendulum held a very important place as a time keeping device. Competent time measurement was a requirement for modern science and the pendulum enabled this to happen (Story behind the science, Web). The pendulum played more than a scientific and technical role in the formation of the modern world. It also indirectly changed cultures and societies through its impact on navigation. Position on the Earth’s surface is given by latitude and longitude. A traveler sailing across the sea must know the coordinates of his present position as well the coordinates of his destination.

Hence the knowledge of position was essential for reliable traveling and trading. Accurate time measurement was long seen as the solution to the problem of longitude determination which had vexed European maritime nations in their efforts to sail beyond Europe’s shores. Treasure fleets from Latin America, trading ships from the Far East were all getting lost and running out of food and water. The pendulum thus played a pivotal role in resolving the longitude problem and thus holds an important place in Physics as well as History.

This thesis will focus on the interesting aspects about the period of a pendulum and its mathematical derivation. According to Hooke’s law, the restoring force of a spring is directly proportional to its displacement. Fig 1: Physical representation of Hooke’s law The above figure shows a spring elongated through a length x. F is the force that wants to drive the spring back to equilibrium. By Hooke’s law, |F| ? |x| F = -kx where k is the spring constant measured in Newton/metre (N/m) Here, the negative sign represents that the direction of F is opposite to that of x.

Moving further on, consider the case