The Relationship between Period, Angular Amplitude and Length of a simple pendulum
Abstract:
With regard to the period of a simple pendulum, the investigation examined to what extent the length of a string and the angular amplitude affect the period, with increasing values of both. By increasing the length of the string, the period was determined an increase in value, alongside an increase in amplitude of the pendulum. The two dependences suggested from the experimental construction and mathematic calculations were the period of small amplitude and large amplitude with the utilisation of the length of the pendulum.
Introduction:
A simple pendulum consists of a point mass 'm', suspended from a fixed point using a mass less ideal string of length 'l', such that it can move forth and back from its mean position.
One complete to and from movement of a pendulum about its mean position is known as an oscillation or vibration. There are two dominant forces acting upon a pendulum bob at all times during the course of its motion: the force of gravity that acts downward upon the bob and the tension force acting upward and towards the pivot point of the pendulum resulting in the pulling of the bob of the pendulum from string.
While the gravity is constant (mass*9.8 N/Kg), the tension force is less predictable and changes as the direction and magnitude differ when the bob swings back and forth. However, the direction of the tension force is always towards the pivot force, resulting it at an angle (directed upwards and to the right) when the pendulum bob swings to the left of its equilibrium position. Oppositely, the tension is directed upwards and to the left when the pendulum bob swings to the right of its equilibrium position. Figure 1 below demonstrates the direction of these two forces at 5 different positions over the course of the pendulum’s path.
Figure 1
During the swings of the pendulum, there’s also the consideration of kinetic and potential energy, which is crucial to acknowledge whilst it’s not the main purpose of performing this particular investigation.
The kinetic energy possessed by an object is the energy produced due to its motion, which depends upon both mass and speed of the object, forming the following equation (kinetic energy (KE) to mass (m) and speed (v)):
KE = ½•m•v2
During the sequence of motions, the faster an object moves, the more kinetic energy that it possesses. In the application of the pendulum bob, the kinetic energy increases as it approaches the equilibrium position and decreases as it moves further away from the equilibrium position, as can be seen in figure 2 that demonstrates the value of KE at special points
Figure 2
On the other hand, the potential energy possessed by an object is the stored energy of different positions. For a simple pendulum, the form of potential energy possessed by it is gravitational potential energy. The amount of gravitational potential energy is dependent upon the mass (m) of the object ...
