HARMONIC OSCILLATIONS IN ONE DIMENSION

OBJECTIVES:

• Write Newton's second law for a particle undergoing simple harmonic motion in one dimension and write the general solution for the resulting differential equation.
• Define the following terms as they relate to a simple harmonic oscillator:
  simple harmonic motion, amplitude, frequency (Hertz), phase constant (or phase angle), angular frequency, period, spring constant, restoring force.
• Given the necessary information about a system oscillating harmonically in one dimension, solve for any of the following:
  1. position as a function of time,
  2. angular frequency,
  3. period,
  4. amplitude,
  5. frequency,
  6. velocity,
  7. acceleration,
  8. mass,
  9. restoring force.

Commentary:

The general equation for simple harmonic motion along the x-axis results from a straightforward application of Newton's second law to a particle of mass m acted on by a force:

F = -kx,

where x is the displacement from equilibrium and k is called the spring constant.

Since the acceleration:

a = dv/dt = d²x/dt²,

Newton's second law becomes:

-kx = m d²x/dt²,

which is called a second-order differential equation because it contains a second derivative.

We can combine the constants k and m by making the substitution:

k/m = ²,

and rewrite this equation as:

d²x/dt² = - ²x .(1)

Your calculus background may not have acquainted you with differential equations; hence, we will discuss them briefly here without getting too fancy or formal.

Equation (1) is not like an algebraic equation for which certain constant values of x satisfy the equality.

The solution of Eq. (1) is a function of time.

Although the function

x(t) = A cos(t + ) = A cos() cos(t) - A sin() sin(t) (2)

can be thought of simply as being arrived at by a very clever guess, it can be shown (by advanced mathematical techniques) to be the most general possible solution of Eq. (1).

Equation (2) can also be written in terms of two new constants B and C as

x(t) = B cos(t) + C sin(t) (3)

(What are the relations among B, C, A, and ?)

The velocity is then

v(t) = dx/dt = -B sin(t) + C cos(t) (4)

These last two equations are especially helpful.

For instance, if you are told that a particle begins its simple harmonic motion from rest at the point x₀, you know that x(0) = x₀ and v(0) = 0; hence, since cos(0) = 1 and sin(0) = 0 you immediately have:

B = x₀ and C = 0.

If the particle starts at the origin (x=0) with velocity v₀, then you can conclude that:

B = 0 and C = v₀.

Look at the equations and check these results for yourself!

If you have a more complicated case in which the particle starts at x₀ with velocity v₀, then you can find B and C yourself, using the same method. Try it!

Once you have found B and C, you can then find A and .

All of the terms listed in the objectives for this section are defined in the readings in the different textbooks on the InfoMall.

Familiarize yourself with the symbols used to identify the parameters of harmonic motion, and recognize that different texts may use different symbols.

For example, you may see  or  (instead of ) used to represent the phase constant. The phase constant determines the initial conditions (displacement and velocity) of the motion. The significance of the angular nature of the phase constant should become somewhat clearer in the next section when we examine the analogy between harmonic motion and uniform circular motion. For now, notice that since is a constant, its value is arbitrary in the general solution of Eq. (1). Even when the initial conditions of the motion (at t = 0) are specified, is only determined to within an integral multiple of 2. That is, if

x = A cos(t + )

describes the motion, then so does

x = A cos(t +  + 2n)

for any integer n= 1, 2,3, ....

The equation:

x = A sin(t +  + /2)

is an equally valid (in fact identical) solution as you can verify for yourself.

Be alert to the difference between frequency, f or , and angular frequency, .

Both can have dimensions of s^-1, but the units of frequency are oscillations/second while those of angular frequency  are radians/second: they are related by:

 = 2 f

Problem:

1. The displacement of an object undergoing simple harmonic motion is given by the equation

x(t) = 3.00 sin(8t + /4)

where x is in meters, t is in seconds and the argument of the sine function is in radians.

1. What is the amplitude of motion?
2. What is the frequency of the motion?
3. What are the position, velocity, and acceleration of the object at t = 0?