Answer to Question #104191 in Trigonometry for tamara

Question

What is “harmonic motion”? How can harmonic motion be described mathematically? Be sure to link correct mathematical terms to correct physics terms – for example, how are the amplitude, period length, and other mathematical properties of harmonic motion waves related to how an object is moving? What is Hooke’s Law, and what does it have to do with harmonic motion? What is a spring constant, and what does it have to do with harmonic motion? Provide a couple of real-life examples of objects that move, roughly, according to simple harmonic motion

Answer 1

Simple harmonic motion, in physics, repetitive movement back and forth through an equilibrium, or central, position, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side. The time interval of each complete vibration is the same, and the force responsible for the motion is always directed toward the equilibrium position and is directly proportional to the distance from it.This motion can be described by the law of sine or cosine

.The offset from the equilibrium position can be calculated by the formula:

x(t) =A*sin(ω*t+φ),

x is the deviation of the oscillating quantity at the current time t from the average value over the period of the value (for example, in kinematics - the offset, deviation of the oscillating point from the equilibrium position);

A is the oscillation amplitude, i.e. the maximum deviation of the fluctuating quantity from the average value for the period over a period, dimension A coincides with dimension x;

ω (radian / s, degree / s) - cyclic frequency, showing how many radians (degrees) the oscillation phase changes in 1 s;

φ is the initial phase of the oscillation, which determines the value of the total phase of the oscillation (and the value of x itself) at time t = 0

The speed :

v(t) =-A*ω*sin(ω*t+φ)

The period of mathematical pendulum :

T=2*π*(l/g)^1/2,

L- length of pendulum, g=9.8 N/m,

Spring pendulum period:

T=2*π*(m/k)^1/2,

m- mass, k is the spring constant

Hooke’s Law:

F=k*x,

F- elastic force,

k is the spring constant, in Newtons per meter (N/m), and x is the displacement of the spring from its equilibrium position. 

Sound and light are examples of oscillatory movements.