To describe simple harmonic
motion with a trigonometric
Simple Harmonic Motion
SIMPLE HARMONIC MOTION: motions that
are periodic in nature such as oscillations of a
pendulum, the orbit of a satellite, the
vibrations of a tuning fork, etc., which can be
described by sine or cosine curves
• A point that moves on a coordinate line is
said to be in simple harmonic motion if its
distance from the origin at time t is given
– d = a sin ωt or d = a cos ωt
• where a and ω are real numbers such that
ω > 0, with amplitude a, period 2π/ω, and
• A ball is bobbing up and down on a spring.
10cm is the maximum distance the ball
moves vertically upward or downward from
its equilibrium point. It takes the ball
t = 4 seconds to move from its maximum
displacement above zero to its maximum
displacement below zero and back again
(assume no friction or resistance)
• When the spring is at equilibrium, d = 0,
when t = 0, so we will use the equation
d = a sin ωt.
• The maximum displacement from zero is 10
and the period is 4.
• Amplitude = 10
• Period = 4, so
• So, d 10 sin t
2 cycle / sec
Port Aransas Pier
The graph indicates the depth of the water off a certain pier in Port
Aransas, Texas, beginning at noon, March 19th.
Note: t = hours after noon on March 19th
y = depth of water in meters
At what time and date is the first high tide? Noon, March 19th
The second high tide? 2 am, March 20th
The first low tide? 7 pm, March 19th
How long does it take the tide to complete one cycle?
Amplitude = 3
14 = 2π/B
B = π/7
y 3cos t
What is the depth of the water at midnight March 19?
Since midnight is 12 hours after noon, let t = 12.
Plug 12 into the equation for the function.
y = 1.870 meters
At what time is the depth of the tide 1-meter?
Graph y1 = equation t = 2.743 and 11.257
Graph y2 = 1 2:45 pm and 11:15 pm
Weight on Spring
A weight is at rest hanging from a spring. It is then pulled down 6
cm and released. The weight oscillates up and down, completing
one cycle every 3 seconds.
Distance above/below resting point, in cm
Amplitude = 6
3 = 2π/B
B = 2π/3
y 6sin ( x 1.5)
Determine the position of the weight at 1.5 seconds.
Let x = 1.5; plug into equation for function.
y = 0 cm (back at original position)
Use the graph to find the time when y = 3.5 for the first time.
Graph y1 = equation you wrote; graph y2 = 3.5.
x = 1.797 seconds
3.5 is the 3.5 cm distance above the original position of the weight.