So, if we instead used the equation Y = sin(2 t), the function will repeat when 2 t gets to 2ℼ, meaning t only has to get up to a value of ℼ before it repeats.
Except it doesn't have to be t that gets to 2ℼ, it just has to be the value inside the parentheses that reaches 2ℼ.
Using the equation Y = sin( t), we could see that the graph begins to repeat once t gets to 2ℼ. It is most common to use radians for these functions, so we'll do that here. Modifying the period of the function can be done by adding a coefficient inside the sine or cosine function, right next to the t. If we noticed that our oscillator has an amplitude of 12 cm, we would simply make the coefficient in front of sine or cosine be 12 cm. For example, the function Y = 3sin( t) has an amplitude of 3 instead of 1. If we put a coefficient in front of the sine or cosine function, the amplitude of the function will be equal to the coefficient we put in. If you need to put one of these into a calculator and graph it, use x rather than t. By convention, we typically use "t" instead of "x" in an equation like this, because we are modeling the position as a function of time. Since there is no requirement that a simple harmonic oscillator have those same values for amplitude and period, we need to modify the equations to match our object's motion. If we were to graph Y = sin( x) and Y = cos( x), we would see that both functions have a maximum value of 1, a minimum value of -1 (so the amplitude of each function is 1), and a period of 2ℼ radians (360 degrees). (SHM is periodic motion caused by a restoring force that is proportional the object's distance from equilibrium. As long as the requirements for simple harmonic motion are met, the motion can be modeled with a sine or cosine function. Since a simple harmonic oscillator (such as a pendulum or a mass on a spring) goes back and forth again and again, we need to model this using a function that does the same thing. For example, we can write equations that describe the position of objects in simple harmonic motion. One way to model something is to write an equation that describes it. The chapter discusses waves that travel on very long springs or strings or across a large expanse of water.Making models to describe the world around us is a huge part of science. The frequency of a wave is how frequently the wave crests pass a given point. A wave pulse is formed when a sharp displacement is given to the end of a spring or string. All types of mechanical wave pulses-whether on springs or strings, on water, or in the air-are characterized by the transfer of motion from particle to particle in the medium in no case, any part of the medium moves any appreciable distance. A type of system that exhibits simple harmonic motion is a simple pendulum.
Many types of elastic materials have the property that when they are stretched or compressed through, there is a restoring force that is proportional to the amount of elongation or compression. It discusses some important features of wave motion. This chapter describes a type of oscillatory particle motion that is related to wave motion-namely, simple harmonic motion. MARION, in Physics in the Modern World (Second Edition), 1981 Publisher Summary The sum of the simple harmonic oscillation of each nucleus, as a function of nucleus position in the lattice and time, produces an overall displacement wave in the crystal lattice, which can be expressed as a sum of normal vibration modes. The displacement of the nuclei from their equilibrium positions, is caused by heating of the crystal lattice or another means of energy exchange, such as neutron collisions.
The nucleus displacement vector u 0 d ( 0 ) at time t = 0 and u 0 d ( t ) for the same nucleus at later time t, as well as displacement vector u l d ′ ( t ) for another nucleus at later time t, can be expressed as the sum of displacements, which arise from a set of normal vibration modes. The interatomic forces between the crystal atoms (ions) are assumed to undergo simple harmonic motion, where the force acting on a displaced nucleus, is proportional to the displacement distance of the nucleus from its equilibrium lattice position. Jay Theodore Cremer Jr., in Advances in Imaging and Electron Physics, 2012 2 The Nucleus Displacement Vector in a Non-Bravais Crystal with Nuclei Vibrations