Simple harmonic motion (SHM) is a unique kind of motion in which the position is a sinusoidal function of time:
Here,is amplitude, is angular frequency, and is the phase of the oscillations. The period of the oscillations can be calculated as . In general, does not have to be a position; it could be any variable describing a system.
Mechanical vibrations and waves are similar to electromagnetic waves. Both mechanical and electromagnetic waves begin with an oscillatory system where vibrations propagate in space. The most basic of all oscillatory phenomena is simple harmonic motion. It can occur in a variety of mechanical systems, as discussed in Vibrations.
Harmonic Motion covers the mathematical description and properties of the simple harmonic motion as applied to a system of a mass on a spring.
Waves surrounding us have such a variety of manifestations – ripples on a surface of water, sound, or light – that it is not easy to realize that they all have the same underlying properties. The main property that distinguishes wave motion from the motion of matter particles is that waves can pass through each other without affecting each other's motion. When more than one wave is present at the same place, they combine. This interference of waves is known as superposition. Waves also demonstrate diffraction, which is the ability to bend around an obstacle. When a wave encounters a boundary between two media, it undergoes reflection (traveling backward) and transmission (also referred to as refraction).
Periodic waves are characterized by their wavelength, which is the distance the wave travels during one period - the time it takes for the oscillation to go through a full cycle. Thus, the relationship between the wavelength and period is, where is the speed of the wave propagation. Alternatively, wavelength can be related to frequency. Frequency is a reciprocal of period, or the number of cycles the wave goes through in one second: . Thus, .
Read Free Waves to review the properties and mathematical description of wave motion.
Simple harmonic motion arises when a restoring force appears in a system when that system is disturbed from equilibrium, and when the force is proportional to and has the opposite direction of the displacement. A typical example of such a system is a point mass attached to the end of a massless spring. In this case, Hooke's law states that the restorative force is, where is the displacement of the mass and is equal to the length by which the spring is stretched or compressed. is a spring constant, determined by the material of the spring and how tightly it is wound. The negative sign indicates that the force is in the direction opposite to the displacement.
Springs and Hooke's Law discusses this system in detail.
Application problems involving simple harmonic motion typically focus on the relationship between the maximum displacement (amplitude) and maximum velocity of a system, and on calculating angular velocity, period, and energy of the oscillations. These are the formulas you should know that pertain to the situations involving SHM:
See the solved examples of application problems in Oscillatory Motion.
This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.
Try to think of the reason why each term is included.