Harmonic Motion

Describe the motion of a rider on a ferris wheel relative to the ground.

Describe the motion of a buoy in an ocean relative to the ocean floor:

Describe the motion of a vibrating horizontal guitar string.

Describe the motion of a piston in a cylinder.

Describe the motion of the mass in a mass-spring system:

Harmonic Motion simple harmonic motion – repetitious motion that would continue without stopping if there were no outside forces on the object

Harmonic Motion “if there were no outside forces on the object”

Harmonic Motion “if there were no outside forces on the object” swinging pendulum

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity mass in a spring-mass system

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity mass in a spring-mass system – gravity, friction

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity mass in a spring-mass system – gravity, friction vibrating guitar string

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity mass in a spring-mass system – gravity, friction vibrating guitar string - tension

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity mass in a spring-mass system – gravity, friction vibrating guitar string - tension floating buoy

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity mass in a spring-mass system – gravity, friction vibrating guitar string - tension floating buoy ferris wheel

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity mass in a spring-mass system – gravity, friction vibrating guitar string - tension floating buoy ferris wheel – mechanical/human

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity mass in a spring-mass system – gravity, friction vibrating guitar string - tension floating buoy ferris wheel – mechanical/human piston in a cylinder

Harmonic Motion “if there were no outside forces on the object” swinging pendulum - gravity mass in a spring-mass system – gravity, friction vibrating guitar string - tension floating buoy ferris wheel – mechanical/human piston in a cylinder – friction/ human

Harmonic Motion We know that the motion of objects with simple harmonic motion can be represented by a sine or cosine function. Here’s why: If you were to trace out the object’s motion over time (or some other x-coordinate quantity), the result would be a sine or cosine function.

Harmonic Motion Imagine attaching a pencil to a guitar string, setting the string into vibration while sliding a sheet of paper to the right underneath the string and pencil. Or imagine attaching a pencil to the outside edge of a circular object and sliding a sheet of paper to the right while the circle was turning counterclockwise.

Harmonic Motion If we know some information about the object’s motion, typically the position at some time and its frequency, then we can write an equation that will model its motion.

Harmonic Motion Problem - solving strategy: make and label a diagram make and label a sine or cosine function graph write an equation to represent the graph answer the question

Harmonic Motion A buoy floats in a harbor at a location where the water depth is 26 ft. As waves pass the buoy it rises and falls. The buoy moves a total of 8 ft from its lowest to its highest point every 12 seconds. Assume the buoy is at its low point at t = 0. Find the height of the buoy after 15 seconds.

Harmonic Motion Sketch a diagram:

Harmonic Motion Sketch a graph to represent the motion:

Harmonic Motion Write a sine or cosine equation: Answer the question:

Harmonic Motion A mass in a spring-mass system is at rest 5 ft from a 10 ft ceiling. The mass is given a push up at time, t = 0. The mass is 2 ft from the ceiling at its highest point and it takes 4 seconds to complete a cycle and return to its starting position. What is the mass’ height 10 seconds after being set into motion?

Harmonic Motion Sketch a diagram:

Harmonic Motion Sketch a graph to represent the motion:

Harmonic Motion Write a sine or cosine equation: Answer the question: