If the spring is extended a distance x, its free end is at x and it exerts a force -k x which is now in the negative x direction:
The constant of proportionality k is called the spring constant, and the relationship
F = - k xis called Hooke's Law. The spring constant clearly has units of force over distance, or kg / s2. Stiffer springs have larger spring constants; a moderately loose spring might have a spring constant of about 10 kg / s2.
The spring provides us with an opportunity for additional physical analogies. During pulsatile flow, arteries expand and contract with the changes in blood pressure. This movement can be modeled using Hooke's Law, where x is replaced by Dr: the variation in the radius of the artery from its equilibrium position. The aorta typically has a spring constant in the neighborhood of 1000, and radial displacements of around 2 mm.
As usual, we are ignoring friction. As a result Hooke's Law describes a conservative force, which can be written as the gradient of a potential energy. Since the gradient introduces an additional power of distance in the denominator, we expect this elastic potential energy to be proportional to x2. Multiplying by our ubiquitous unitless factor of 1/2, we find that
U = k x2 / 2.
Small Oscillations
The spring is a nearly universal model in physics. This is true in part because only it and the 1/r2 force can describe exactly periodic phenomena. As a result, we have the mathematical tools necessary to completely analyze such systems. But the primary utility of the spring goes back to our original observation: it bounces.
Consider a small mass on the free end of a spring. If we displace the mass slightly away from equilibrium, the elastic force will accelerate it back toward its equilibrium position. When it reaches equilibrium, however, it has a nonzero momentum and overshoots that position. The elastic force now accelerates the mass in the opposite direction, back toward the equilibrium position. This periodic motion is called oscillation.
If we combine Hooke's Law with Newton's Law, we find that
m a = - k x,
or
a = (-k / m) x.
In words, this means that the rate of change of the rate of change of position is proportional to the position. Graphically, we can try to picture the position as an oscillatory function of time: perhaps a sine function:
At each point on that graph, the slope gives us the velocity of the mass at that time. The velocity graph must also be an oscillatory function of time:
and the slope at any point on this graph gives us the acceleration of the mass. Clearly, the graph of acceleration versus time must also be oscillatory, and to satisfy our equation, every point on it must be proportional to the value of the position at that time, but reflected about the x axis because of the minus sign:
Both the sine and cosine functions have this property: the slope of the slope of the function at any point is proportional to the negative of the function. To be specific, our simple harmonic oscillator could be described by either
Notice what happens when we plot the position in black, the square of the velocity, which is proportional to the mass' kinetic energy, in red, and the square of the position, which is proportional to its potential energy, in blue:
The kinetic energy is always a maximum at the equilibrium position, where the potential energy is zero, and the potential energy is always a maximum at the extremes of the oscillation, where the velocity (and kinetic energy) is zero. Conservation of energy then tells us that the total energy of the oscillator is just the potential energy at the maximum displacement from equilibrium. This displacement is called the amplitude A, and the total energy is
-A w sin (w t),
= k A2 (sin (w t)2 + cos (w t)2) / 2
= k A2 / 2
The arguments of trigonometric functions must always be unitless. The variable w (which in rotational motion was used to denote the angular velocity) is called the angular frequency and has units of 1 / s, so that the argument of the cosine function is indeed unitless. Dividing w by 2p we find the frequency n (the Greek letter nu) which is the number of oscillations or cycles per second from the maximum amplitude through zero to the minimum amplitude and back to the maximum again (each of the graphs above was one cycle). The inverse of the frequency is the period T, which is the time in seconds for one oscillation (and is therefore always positive).
In general, the argument of an oscillatory function is called the phase. The phase can also be a function of x: k x, where now k is called the wave number and has units of 1 / m. The wavelength l (the Greek letter lambda) is 2p / k and is analogous to the period. Like the period, the wavelength is always positive.
The phase can also include a term which is a unitless number (denoted by the Greek letter delta), so in its most general form the phase is written
The following applet will help you to visualize how the sine and cosine functions depend on the wave number and phase angle (we have
set w to zero to provide a static display). It plots the sine or cosine of the phase from -2p
to 4p:
Pay close attention to the values of the phase for which the functions are a maximum, a minimum, or zero. Note also the phase differences
between consecutive zeroes, maxima or minima.
Compare the effect of various phase values to what happens to a function like y(x) = x2 when you
replace x by (a x + b).
Now you are ready to do some problems.
This applet is designed to help you use the phase to understand the behavior of the oscillator as a function of time or position. When you answer the question correctly, the program will graph the function so that you can visualize the effect of the phase on oscillation. Note that all answers will be positive and within one cycle of the origin, so you may need to add or subtract one or more half periods (or wavelengths) so that the desired point on the oscillation is in that region.
Note that the answers you computed for the second applet differ by an integer times T / 4 (or l / 4).
The next section introduces us to waves, and discusses the effects of boundary conditions and superposition on standing waves.
©2007, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.
Please send comments or suggestions to the author.
x(t) = sin (w t)
or
x(t) = cos (w t),
where
w = (k / m)1/2.
In this case, we choose the cosine function, because at time t = 0 the mass was displaced a small distance from the origin; since sin (t) is zero at time zero, only the cosine can describe these oscillations.
k A2 / 2.
In fact, the position and velocity of this oscillator are
A cos (w t) and
so that the sum of the kinetic and potential energies is
m (-A w sin (w t))2 / 2 +
k (A cos (w t))2 / 2,
at every point along its trajectory.
= m w2 A2 sin (w t)2 / 2 +
k A2 cos (w t)2 / 2
k x +- w t +- d.
d is called the phase angle, and effectively allows us to specify the relative starting point of the oscillator at time zero. By experimenting with various values of d (ie., 0, p/2, p, 3p/2, 2p), we see that we can produce oscillations which have any given initial value (between -A and A) at time zero.