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Oscillations and Simple Harmonic Motion

Simple Harmonic Motion

Problems

Problems

Having established the basics of oscillations, we now turn to the special case of simple harmonic motion. We will describe the conditions of a simple harmonic oscillator, derive its resultant motion, and finally derive the energy of such a system.

The Simple Harmonic Oscillator

Of all the different types of oscillating systems, the simplest, mathematically speaking, is that of harmonic oscillations. The motion of such systems can be described using sine and cosine functions, as we shall derive later. For now, however, we simply define simple harmonic motion, and describe the force involved in such oscillation.

To develop the idea of a harmonic oscillator we will use the most common example of harmonic oscillation: a mass on a spring. For a given spring with constant k , the spring always puts a force on the mass to return it to the equilibrium position. Recall also that the magnitude of this force is always given by:

F(x) = - kx    

where the equilibrium point is denoted by x = 0 . In other words, the more the spring is stretched or compressed, the harder the spring pushes to return the block to its equilibrium position. This equation is only valid if there are no other forces acting on the block. If there is friction between the block and the ground, or air resistance, the motion is not simple harmonic, and the force on the block cannot be described by the above equation.

Though the spring is the most common example of simple harmonic motion, a pendulum can be approximated by simple harmonic motion, and the torsional oscillator obeys simple harmonic motion. Both of these examples will be examined in depth in Applications of Simple Harmonic Motion.

Simple Harmonic Motion

>From our concept of a simple harmonic oscillator we can derive rules for the motion of such a system. We start with our basic force formula, F = - kx . Using Newton's Second Law, we can substitute for force in terms of acceleration:

ma = - kx

Here we have a direct relation between position and acceleration. For you calculus types, the above equation is a differential equation, and can be solved quite easily. Note: The following derivation is not important for a non- calculus based course, but allows us to fully describe the motion of a simple harmonic oscillator.

Deriving the Equation for Simple Harmonic Motion

Rearranging our equation in terms of derivatives, we see that:

m = - kx

or

+ x = 0    

Let us interpret this equation. The second derivative of a function of x plus the function itself (times a constant) is equal to zero. Thus the second derivative of our function must have the same form as the function itself. What readily comes to mind is the sine and cosine function. Let us come up with a trial solution to our differential equation, and see if it works.

As a tentative solution, we write:

x = a cos(bt)

where a and b are constants. Differentiating this equation, we see that

= - ab sin(bt)

and

= - ab 2cos(bt)

Plugging this into our original differential equation, we see that:

- ab 2cos(bt) + a cos(bt) = 0

It is clear that, if b 2 = , then the equation is satisfied. Thus the equation governing simple harmonic oscillation is:

simple

x = a cos t    

The Equation for Simple Harmonic Motion

From the equation for simple harmonic motion we can tell a lot about the motion of a harmonic system. First of all, x is maximum when the cosine function is equal to 1, or when x = a . Thus a in this equation is the amplitude of oscillation, which we have already denoted by x m . Secondly, we can find the period of oscillation of the system. At t = 0 , x = x m . Also, at t = 2Π , x = x m . Since both these instances have the same position, the time between the two gives us our period of oscillation. Thus:

T = 2Π    

and

ν = =    

finally,

σ = 2Πν =    

Note that the values of period and frequency depend only on the mass of the block and the spring constant. No matter what initial displacement is given to the block, it will oscillate at the same frequency. This concept is important. A block with a small displacement will move with slower velocity, but with the same frequency as a block with a large displacement.

Notice also that our value for σ is the same as what we called the constant b in our original equation. So now we know that a = x m and b = σ . In addition we can take the time derivative of our equation to generate a full set of equations for simple harmonic motion:


x = x mcos(σt)  
v = - σx msin(σt)  
a = - σ 2 x mcos(σt)  

Thus we have derived equations for the motion of a given simple harmonic system.

Energy of a Simple Harmonic Oscillator

Consider a simple harmonic oscillator completing one cycle. In the jargon of conservative vs. nonconservative forces (see Conservation of Energy the oscillator has completed a closed loop, and returns to its initial position with the same energy it began with. Thus the simple harmonic oscillator is a conservative system. Since the velocity of the oscillator does change, however, there must be an expression for the potential energy of the system, such that the total energy of the system is constant.

We already know the kinetic energy of the system at any given time:


K = mv 2  
  = m(- σx msin(σt))2  
  = kx m 2sin2(σt)  

The kinetic energy has a maximum value when the potential energy is zero, and sin(σt) = 1 . Thus K max = kx m . Since the potential energy is zero at this point, this value must give the total energy of the system. Thus, at any time, we can state that:


E = U + K  
kx m 2 = U + kx m 2sin2(σt)  

Solving for U:

U = kx m 2(1 - sin2(σt))

Recall that sin2 a + cos2 a = 1 . We can thus substitute:

U = kx m 2cos2(σt)

However, we also know that x = x mcos(σt) for any simple harmonic oscillation. Using this knowledge we can further simplify our equation for potential energy:

simplify

U = kx 2    

With this equation we have an expression for the potential energy of a simple harmonic oscillator given a displacement from equilibrium. When examined practically, this equation makes sense. Consider our example of a spring. When the spring is stretched or compressed a large amount (i.e. the block on the spring has a large magnitude for x ), there is a great deal of energy stored in those springs. As the spring relaxes and accelerates the block this potential energy gets converted to kinetic energy. Shown below are three positions of the oscillating spring, and the energies associated with each position.
Figure %: An oscillating block a) at equilibrium with only kinetic energy b) at maximum displacement with only potential energy c) partially displaced with both potential and kinetic energy. Note that the total energy in each case is kx m 2 .

This SparkNote introducing oscillation and simple harmonic motion involved a great deal of mathematics and theoretical calculations. In the next SparkNote we explore oscillations on a more practical level, examining real physical situations and various types of oscillators.

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