# 5. Applications of Trigonometric Graphs

by M. Bourne

Oscilloscope output - Filter modulation [Image source: Mikael Altemark]

## 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:

### Need Graph Paper?

d = R cos ωt

d = R sin (ωt + α)

d = R cos (ωt + α)

### Example 1

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

e = 0.014 cos(2πft),

where e is in volts and f is in Hz.

Draw 2 cycles of e for f = 950\ "kHz".

## 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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