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If you drag in a spring and drop, it will swing back and forth. In this situation Hookes law plays an important role. The Law is used to describe the spring force. It is defined as: F = , it says that the spring force F is proportional with the displace x. Using Newton's second law, you can set the following second-order differential equations: y’’ = -ky, the complete solution to this is given by: Asin( t) + Bcos( t). This motion is periodic, i.e. each time it increases by 2π, a period is gone.
The harmonic event is an ideal borderline case. There will always be friction that causes the movement to stop at a time. By the muted event, there will be added a spare part on the homogeneous equation, so it will look like this: . The solutions to this depends on the so-called character equation, which is as follows: R2 + R + = 0. The solutions to the subdued event, dependents on the sign in front of discriminant to the character equation. The solutions to the subdued case has an exponential decreasing amplitude. We will make an experiment which shows this. In connection with it, we will define the following differential equations: ., and then we want to find all the solutions: .