Now that we have established the theory and equations behind harmonic motion, we will examine various physical situations in which objects move in simple harmonic motion. Previously, we worked with a mass-spring system, and will examine other harmonic oscillators in a similar manner. Finally, after establishing these applications, we can examine the similarity between simple harmonic motion and uniform circular motion.
Consider a circular disk suspended from a wire fixed to a ceiling. If the disk is rotated, the wire will twist. When the disk is released, the twisted wire exerts a restoring force on the disk, causing it to rotate past its equilibrium point, twisting the wire the other direction, as shown below. This system is called a torsional oscillator.
τ = - κθ |
If we substitute m for I , k for κ , and x for θ we can see that this is the exact same differential equation we had for our spring system. Thus we may skip to the final solution, describing the angular displacement of the disk as a function of time:
θ = θ _{m}cos(σt) |
From our expression for angular frequency we can derive that
T = 2Π |
From our examination of the torsional oscillator we have derived that its motion is simple harmonic. This oscillator can almost be seen as the rotational analogue of the mass-spring system: just as with the mass-spring we substituted θ for x , I for m and κ for k . Not all simple harmonic oscillators have such close correlation.
Another common oscillation is that of the simple pendulum. The classic pendulum consists of a particle suspended from a light cord. When the particle is pulled to one side and released, it swings back past the equilibrium point and oscillates between two maximum angular displacements. It is clear that the motion is periodic--we want to see if it is simple harmonic.
We do so by drawing a free body diagram and examining the forces on the pendulum at any given time.
F = - mg sinθ |
This equation does predict simple harmonic motion, as force is proportional to angular displacement. We can simplify by noticing that the linear displacement of the particle corresponding to an angle of θ is given by x = Lθ . Substituting this in, we see that:
F = - mg = - x |
pendulum
T = 2Π = 2Π |
The torsional oscillator and the pendulum are two easy examples of simple harmonic motion. This type of motion, described by the same equations we have derived, comes up in molecular theory, electricity and magnetism, and even astronomy. The same method we applied in this section can be applied to any situation in which harmonic motion is involved.
Through our study of simple harmonic oscillations we have used sine and cosine functions, and talked about angular frequency. It seems natural that there should be some connection between simple harmonic motion and uniform circular motion. In fact, there is an astonishingly simple connection that can be easily seen.
Consider a particle traveling in a circle of radius R centered about the origin, shown below:
However, if the particle is traveling with a constant angular velocity σ , then we can express θ as: θ = σt . In addition, the maximum value that x can take is at the point (R,0), so we can state that x _{m} = R . Substituting these expressions into our equation,
x = x _{m}cos(σt) |
Simple harmonic motion can be seen as the projection of a particle in uniform circular motion onto the diameter of the circle.
This is an astonishing statement. We can see this relation through the following example. Place a mass on a spring such that its equilibrium point is at the point x = 0 . Displace the mass until it is at the point (R,0). At the same time that you release the mass, set a particle in uniform circular motion from the point (R,0). If the two systems have the same value for σ , then the x coordinate of the position of the mass on the spring and the particle will be exactly the same. This relation is a powerful application of the concepts of simple harmonic motion, and serves to increase our understanding about oscillations.