Trigonometry/Applications and Models

Simple harmonic motion


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


 * $$x = A \sin \left(\omega t + \phi\right)$$

or


 * $$x = c_{1} \cos\left(\omega t\right) + c_{2} \sin\left(\omega t\right)$$
 * where c1 = A sin &phi; and c2 = A cos &phi;.

In the above functions, A is the amplitude of the motion, &omega; is the angular velocity, and &phi; is the phase.

The velocity of an object in SHM is
 * $$v = A \omega \cos \left(\omega t + \phi\right)$$

The acceleration is
 * $$a = -A \omega^2 \sin \left(\omega t + \phi\right) = -\omega^2 x$$

An alternative definition of harmonic motion is motion such that
 * $$\displaystyle a = -\omega^2 x$$

Springs and Hooke's Law
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
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:


 * $$a = -\frac{k}{m}x$$

The calculus definition of acceleration gives us


 * $$x'' = -\frac{k}{m}x$$


 * $$x'' + \frac{k}{m}x = 0$$

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


 * $$x = c_{1} \cos\left(\sqrt{\frac{k}{m}}t\right) + c_{2} \sin\left(\sqrt{\frac{k}{m}}t\right)$$   (2)

with an independent variable t for time.

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


 * $$\sin \phi = \frac{c_{1}}{\sqrt{c_{1}^{2} + c_{2}^{2}}}$$
 * $$\cos \phi = \frac{c_{2}}{\sqrt{c_{1}^{2} + c_{2}^{2}}}$$

and


 * $$c_{1} = \sqrt{c_{1}^{2} + c_{2}^{2}} \sin \phi$$
 * $$c_{2} = \sqrt{c_{1}^{2} + c_{2}^{2}} \cos \phi$$

Substituting into (2), we get


 * $$x = \sqrt{c_{1}^{2} + c_{2}^{2}} \sin \phi \cos\left(\sqrt{\frac{k}{m}}t\right) + \sqrt{c_{1}^{2} + c_{2}^{2}} \cos \phi \sin\left(\sqrt{\frac{k}{m}}t\right)$$

Using a trigonometric identity, we get:


 * $$x = \sqrt{c_{1}^{2} + c_{2}^{2}} \left[\sin \left(\phi + \sqrt{\frac{k}{m}}t\right) + \sin \left(\phi - \sqrt{\frac{k}{m}}t\right)\right] + \sqrt{c_{1}^{2} + c_{2}^{2}} \left[\sin \left(\sqrt{\frac{k}{m}}t + \phi\right) + \sin \left(\sqrt{\frac{k}{m}}t - \phi\right)\right]$$


 * $$x = \sqrt{c_{1}^{2} + c_{2}^{2}} \sin \left(\sqrt{\frac{k}{m}}t + \phi\right)$$   (3)

Let $$A = \sqrt{c_{1}^{2} + c_{2}^{2}}$$ and $$\omega^{2} = \frac{k}{m}$$. Substituting this into (3) gives


 * $$x = A \sin \left(\omega t + \phi\right)$$

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