Part I - To study the motion of a vibrating spring (Simple Harmonic Motion).
Part II - To determine the period, amplitude, and acceleration due to gravity of a simple pendulum. .
Theory (Part I).
Simple harmonic motion occurs when there is a force that increases linearly with distance as the body moves away from an equilibrium position. The force is always directed toward the equilibrium position. As determined by Robert Hooke in the 17th century, the equation for this force is given by: . This equation explains the proportional relationship between the force enacted upon the spring and the length of which it was displaced or stretched. Where x is the displacement from the equilibrium position and k is the constant of proportionality or the spring constant. The negative sign would be the force that is opposite of the directions of the force applied. The mass would be replaced with the amplitude of the spring's displacement because it the mass would be equals the displacement of the weight due to gravity. We already know that and that the acceleration of spring therefore can be postulated to be a constant. By replacing Fs , in Hooke's equation, with ma since , Newton's Second Law of Motion, gives ma = -kx or . This new equation is a general characteristic of simple harmonic motion where acceleration is proportional to the displacement and oppositely directed. .
Hooke's law is valid only under certain conditions. Specifically, the displacement must not be excessively large. If the displacement (stretching or compression of the spring) exceeds a certain limit (which is different for each spring), Hooke's equation will not work. For stretching forces, there is a limit to how far you can stretch the spring before its elastic limit is exceeded and it is permanently deformed and it will no longer return the mass to the original equilibrium position. As long as these two limits are not exceeded, a mass suspended from a spring will, if displaced from its equilibrium, exhibit simple harmonic motion about the equilibrium point and the restoring force will be accurately described by Hooke's equation.
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