Class: Pre-Calculus
Grade Level: 12th
Unit: 6-7 Simple Harmonic Motion
Teacher: Ms. Jamie Davis
Objectives
Students will be able to apply knowledge of trigonometric functions.
Students will be able to evaluate problems involving simple harmonic motion.
Iowa Core Curriculum-Subject Area Standard
Connections to functions and modeling
creating equations
model periodic phenomena with trig functions
define trig ratios and solve problems involving right triangles
21st Century Skill(s)
Monitor, define, prioritize, and complete tasks without direct oversight
utilize time and manage workload efficiently
evaluate information critically and competently
Essential Question
Can students apply the knowledge learned about trigonometric functions in order to solve
for simple harmonic motions?
Anticipatory Set
1) evaluate y= sinx where -90≤x≤90;
-1≤y≤1
2) y= arctan x where x is all real numbers;
-90≤y≤90
Write the equation for the inverse of each function
3) y= arcos x
y=cosx
4) y= sinx
y= arcsinx
5) give an example showing that tan-1(tanx) ≠x if x<-90 or x>90
answers may vary. If x= 135, tan-1(tan135)= tan-1(-1)= -45
Teaching: Activities
Simple harmonic motion- rhythmic movement of an object where friction and other
factors do not play in affect (are ignored)
ex: buoy in waters
vibrations of guitar strings
pistons of an engine
pendulum swinging back and forth, etc.
Relative to physics especially
Angular velocity- number of degrees/radians that a point on the edge of an object moves
in a unit of time when turning
Wheel example:
wheel turns counterclockwise at constant angular
velocity
P is point on the rim of the wheel
center of the wheel at (0,0) of a circle
Angular velocity is constant, so the measure of angle
𝜃as a function of time is 𝜃= kt
k= constant
t= time
Let A be the measure of the radius of the wheel
substitute kt for 𝜃
So, cos𝜃= cos(kt)= x/a
sin𝜃= sin (kt)=y/A
solving for x and y creates equations for horizontal and vertical motion of point F if
initial position was at (A,0)
x= A cos kt
y= A sin kt
assumes that 𝜃=0 at time t=0
if 𝜃 measure c radians at t=0, then 𝜃 = kt +c
know amplitude of each is above |A|, the period is 2pi/k (or 360/k) and the phase shift is c/k (phase shift in terms of t; angular phase shift is –c)
frequency= inverse of period
these equations help model simple harmonic motion, but modified for the individual
problems
ex 1
write two possible equations to describe the motion of the buoy
given: buoy moves a total of 6 feet from its high point to low point and returns back to
high point every 10 seconds; assume t=0 at its high point
-find the amplitude, period, and phase shift of the function
amplitude, |A|, is 6/2 or 3 feet
period is 10seconds, so 10= 2π/k and k= π/5
initial state x= A cos kt
thus, x= 3 cos (πt/5)
alternate equation using y= A sin (kt + c)
let’s say that 2.5 seconds before a new cycle begins, or ¼ of a complete cycle, the buoy is
at 0 feet…
phase shift= -2.5 seconds (negative, because before a new cycle)
since -2.5 = -c/k and k = π/5, c= π/2
thus y= 3 sin ( (πt/5)+( π/2))
draw the graph of the motion of the buoy in 30 seconds
y = 3 sin ( (πt/5)+( π/2))
frequency- applied to problems involving sounds waves or electricity
= number of cycles per unit of time
= reciprocal of the period
ex: frequency in buoy example is 1/10 or 1/10 of a cycle per second
ex2: lasso spinning with diameter= 6 feet
loop spun 50 times/minute
lowest point= 6 inches above the ground
write equation to describe the highest point above ground after t seconds
freq= 50 rev/min or 50 rpm
one revolution every 60/50 or 1.2 seconds
period= 1.2 = 2π/k and k = 5 π/3 radians/second
starts at 6in above ground, where t=0
so, let c=0 at t=0
use center of rotation as origin of coordinate system
so, radius = 3 feet (6 feet= diameter), which makes coordinate starts at (0,-3)
h= A cos (kt + c)
= -3cos ((5 π/3)t+0)
= -3cos (5 π/3)t
--gives position of the point relative to center of rotation
which is 3 ft 6in (3.5 feet) above the ground
so, equation for height of the point relative to the ground is h= -3cos (5 π/3)t + 3.5
Closure
Have them write down one thing that has a simple harmonic motion in which this applies
to
Independent Practice
Page 349 # 14-24 all
Assessment
Find amplitude, period, frequency, and phase shift:
𝜋𝑡
𝜋
4. s= 7cos ( 2 + 4 )
= 7, 4,1/4, -1/2
5. V= -8 sin (4t- π)
= 8, pi/2, 2/pi, pi/4
6. E= 120 sin 100πt
= 120, 1/50, 50, 0
3𝜋
7. y= -4 sin (3t+ 4 )
= 4, 2pi/3, 3/2pi, -pi/4
Write equations with phase shift=0 under given circumstances
8. Initial position= -3, amplitude 3, period= 2
y= -3cos(pi*t)
9. Initial pos= 0, amp=5, period=2
y=5sin (pi*t)
10. Initial pos= 0, amp=0, period=10
y=7sin (pi*t/5)
11. Initial pos= 0, amp=7, period=10
y=cos (pi*t/5)
12. Y-coordinate of P on rim of wheel 12 ft in diameter that is turning at 10rpm; let P be
(6,0) at t=0
y= 6sin (pi*t/3)
13. Ken observes a buoy at the bottom of a wave bobbing up and down a total distance of
10 feet. The buoy completes a full cycle every 12 seconds.
y= -5cos (pi*t/6)
Materials
calculator
Duration
warm-up- 10
teaching- 40
closure- 5
assessment- 20
homework- remaining time