Trigonometry/Applications and Models
Simple harmonic motion[edit]
Simple harmonic motion (SHM) is the motion of an object which can be modeled by the following function:
or
- where c_{1} = A sin φ and c_{2} = 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 c_{1} and c_{2} be the legs of a right triangle, with angle φ adjacent to c_{2}, 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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