Simple Harmonic Motion Equations

The motion equation for simple harmonic motion contains a complete description of the motion, and other parameters of the motion can be calculated from it.

The velocity and acceleration are given by The total energy for an undamped oscillator is the sum of its kinetic energy and potential energy, which is constant at

Potential Energy

Potential energy is energy which results from position or configuration. An object may have the capacity for doing work as a result of its position in a gravitational field (gravitational potential energy), an electric field (electric potential energy), or a magnetic field (magnetic potential energy). It may have elastic potential energy as a result of a stretched spring or other elastic deformation.

Elastic Potential Energy

Elastic potential energy is Potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring. It is equal to the work done to stretch the spring, which depends upon the spring constant k as well as the distance stretched. According to Hooke's law, the force required to stretch the spring will be directly proportional to the amount of stretch.

Since the force has the form F = -kx then the work done to stretch the spring a distance x is

Spring Potential Energy

Since the change in Potential energy of an object between two positions is equal to the work that must be done to move the object from one point to the other, the calculation of potential energy is equivalent to calculating the work. Since the force required to stretch a spring changes with distance, the calculation of the work involves an integral.

The work can also be visualized as the area under the force curve:

Potential Energy Derivative

If the potential energy function U is known, the force at any point can be obtained by taking the derivative of the potential.

Potential Energy Integral

If the force is known, and is a conservative force, then the potential energy can be obtained by integrating the force.