Simple harmonic motionEdit
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 LawEdit
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 derivationEdit
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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