5. Applications of Trigonometric Graphs
by M. Bourne
Simple Harmonic Motion
Any object moving with constant angular velocity or moving up and down with a regular motion can be described in terms of SIMPLE HARMONIC MOTION.
The displacement, d, of an object moving with SHM, is given by:
d = R sin ωt
where R is the radius of the rotating object and ω is the angular velocity of the object.
For an animation of this concept, go back to: sin animation.
NOTE: We may need to use one the following, depending on the situation:
d = R cos ωt
d = R sin (ωt + α)
d = R cos (ωt + α)
EXAMPLE 1:
Need Graph Paper?
A point on a cam is 8.30 cm from the centre of rotation. Sketch 2 cycles of d as a function of t, given that d = 0 cm when t = 0 s and ω = 3.20 rad/s.
EXAMPLE 2:
The voltage of an alternating current circuit is given by
e = E cos(ωt + α).
Sketch 2 cycles of the voltage as a function of time if
E = 80 V, ω = 377 rad/s and α = π/2.
EXAMPLE 3:
The signal received by a radio is given by
e = 0.014 cos(2πft),
where e is in volts and f is in Hz.
Draw 2 cycles of e for f = 950 kHz.
Here's how it looks in LiveMath.
Angular Velocity
Another important result in this section is:
The angular velocity ω (in radians per second) of a rotating object, is given by:
ω = 2πf
where f is the frequency of the motion, in cycles per second.
Exercises
1. A satellite is orbiting the earth so that its displacement D north of the equator is given by
D = A sin(ωt + α).
Sketch 2 cycles of D as a function of t if
A = 500 km, ω = 3.60 rad/hr and α = 0.
2. Using e = E cos(ωt + α), sketch 2 cycles of the voltage as a function of time if
E = 170 V, ω = 120π rad/s and α = π/3.
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