simple harmonic motion V2.notebook
November 15, 2013
Applications and Models of Trigonometric Functions
Example 1
From the time a small airplane is 100 feet high and 1600 ground feet from its landing runway, the plane descends in a straight line to the runway. Determine the plane's angle of descent.
In Surveying and navigation, directions are generally given in terms of bearings.
A Nautical bearing measures the acute angle that a path or line of sight makes with a fixed north­south line. In Air navigation, bearings are measured in degrees clockwise from north.
simple harmonic motion V2.notebook
November 15, 2013
Example 2
A sailboat leaves a pier and heads due west at 8 knots. After 15 minutes the sailboat tacks, changing course to N 16o W at 10 knots. Find the sailboat's bearing and distance from the pier after 12 minutes on this course.
Simple Harmonic Motion
• A wide variety of problems involve the rhytmic motion of an object.
• Examples:
­ motion of a buoy
­ vibrations of a guitar string
­ pistons of an engine
­ pendulum moving back and forth
• When friction and other factors affecting such motion are ignored, this movement is called Simple Harmonic Motion .
simple harmonic motion V2.notebook
November 15, 2013
Frequency = number of cycles per second
Find a model for simple harmonic motion that satisfies the specified conditions.
Displacement 0 Amplitude 4 cm Period 6 sec What is the frequency of this harmonic motion?
simple harmonic motion V2.notebook
November 15, 2013
Given the equation for simple harmonic motion:
a) find the maximum displacement.
b) find the frequency.
c) Find the value of d when t = 4.
d) Find the least positive value of t for which d = 0
A buoy oscillates in simple harmonic moon as waves go past. It is noted that the buoy moves a total of 3.5 feet from its low point to its high point every 10 seconds. Write an equaon that describes the moon of the buoy if its high point is at t = 0.
High Point
3.5 .
Equilibrium
Low point
simple harmonic motion V2.notebook
November 15, 2013
The motion of a floating leaf can be described by the equation:
where y is measured in centimeters and t is measured in seconds. Find the maximum displacement and the frequency of the leaf.
A weight on a spring bounces a maximum of 8 inches above and below its equibrium point.
The time for one complete cycle is 2 seconds.
a) write two possible equations to describe the motion of the weight
b) draw a graph to show the motion of the weight in 4 seconds.
simple harmonic motion V2.notebook
November 15, 2013
A ball that is bobbing up and down on the end of a spring has a maximum displacement of 3 inches. Its moon (in ideal condions) is modeled by (t > 0),
where y is measured in feet and t is the me in seconds.
A) Graph the funcon.
B) What is the period of the oscillaons?
C) Determine the first me the weight passes the point of equilibrium (y = 0).
A guitar string is fixed at both ends and plucked in the middle. Waves travel down the string toward either end and are reflected. The reflected wave from the right side can be described by the funcon:
y = sin [π(x – t)], and the wave from the le side can be described by:
y = sin [(2π)(x + t)],
where t is the me in seconds.
Draw a sketch of the wave paern on the string aer 4 seconds.