Trigonometry/Applications and Models

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Simple harmonic motion[edit]

Simple harmonic motion. Notice that the position of the dot matches that of the sine wave.

Simple harmonic motion (SHM) is the motion of an object which can be modeled by the following function:

or

where c1 = A sin φ and c2 = A cos φ.

In the above functions, A is the amplitude of the motion, ω is the angular velocity, and φ is the phase.

The velocity of an object in SHM is

The acceleration is

An alternative definition of harmonic motion is motion such that

Springs and Hooke's Law[edit]

An application of this is the motion of a weight hanging on a spring. The motion of a spring can be modeled approximately by Hooke's law:

F = -kx

where F is the force the spring exerts, x is the extension in meters of the spring, and k is a constant characterizing the spring's 'stiffness' hence the name 'stiffness constant'.

Calculus-based derivation[edit]

From Newton's laws we know that F = ma where m is the mass of the weight, and a is its acceleration. Substituting this into Hooke's Law, we get

ma = -kx

Dividing through by m:

The calculus definition of acceleration gives us

Thus we have a second-order differential equation. Solving it gives us

(2)

with an independent variable t for time.


We can change this equation into a simpler form. By lettting c1 and c2 be the legs of a right triangle, with angle φ adjacent to c2, we get

and

Substituting into (2), we get

Using a trigonometric identity, we get:

(3)

Let and . Substituting this into (3) gives


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