A-level Physics/Forces, Fields and Energy/Oscillations

If you observe the motion of a pendulum, a child on a swing, or a speaker cone playing a low frequency sound, you will notice that in each case, there is movement backwards and forwards of the same distance from a center point, or in other words, a vibration. These objects that vibrate are said to oscillate.

Free oscillations
When an object is in free oscillation, it vibrates at its natural frequency. For example, if you strike a tuning fork, it will begin to vibrate for some time after you struck it, or if you hit a pendulum, it will always oscillate at the same frequency no matter how hard you hit it. All oscillating objects have a natural frequency, at which they will vibrate at once they have been moved from the equilibrium position.

Forced oscillations
Imagine a building in an earthquake. The ground is moving side to side, and the building (assuming that it is strong enough to not be completely destroyed by the forces) will be moving side to side with the ground. In this case, this oscillation is not the buildings natural frequency, but it is being forced to vibrate with the ground. This is a forced oscillation.

Examples of oscillating systems

 * A mass that is held up with a spring
 * A pendulum
 * A string of a guitar

Describing oscillations
Oscillations can be shown on a displacement-time graph, like this:



Notice that the curves are smooth. This is because the object slows down before changing direction, instead of bouncing back and forth, which is what a graph with straight lines and sharp corners would describe. Movement that has a displacement-time graph with curved lines like the one above, is called sinusoidal motion.

The graph can show us the differences between several oscillating systems. For an oscillating system, the graph shows us:


 * The displacement at a given point in time,
 * The amplitude,
 * The period and,
 * The frequency

Displacement
The displacement at a certain point in time is the distance of the object away from the centre point. The displacement is 0 at the centre, at its maximum at one end (usually on the right when right is taken as positive), and at its greatest negative value on the opposite end (usually left but, again, only when right is taken as positive). Displacement is given the symbol s or x.

Amplitude
The amplitude is the greatest displacement of an oscillating object. It is measured from the center point to one of the maximum points of displacement. The amplitude can increase or decrease with time. Amplitude is represented by the symbol A

Period and frequency
period is the time taken for a single oscillation. the frequency is the number of oscillations per second

Simple harmonic motion
A body executes simple harmonic motion if its acceleration is proportional to its displacement from a fixed point, and is always in the direction of that point.

To explore simple harmonic motion (SHM) let's take the example of a spring with a mass in the absence of gravity (interestingly, you get SHM even with gravity present). If this is our ideal spring, the force is kx where k is a measure of the stiffness of the spring and x is the displacement. The force is toward the origin if that is the equilibrium position of the spring, so we write -kx to remind ourselves of that. Now, Newton's second law becomes

$$m \frac{d^2 x}{dt^2} = -kx$$.

This differential equation is easy enough to solve, and the answer is $$A cos( \omega_0 t + \phi)$$ where A and $$\phi$$ are arbitrary constants and $$\omega_0 = \sqrt{k/m}$$. It does not really matter how we got the solution, because we are physicists, not mathematicians. This is the answer we are expecting, so we try it, and lo and behold, it works. If you do not believe me, substitute it in. Moreover, this is the complete solution, and you will just have to believe me on that because it is slightly more difficult to prove.

Without loss of generality, we will take $$\phi$$, also called the phase shift, to be zero (if you are concerned about this, we are just defining where t=0 is).

Now, a remarkable thing we recognize about the solution is that the frequency (radians per second), $$\omega_0$$ is independent of A. That is, no matter how big the oscillations are, the frequency is the same. A pendulum approximately undergoes SHM, so this is why they are used in clocks, the amplitude doesn't affect the period! By the way, we have added the subscript zero to omega because we are going to have some other omegas soon.

Some terms to remember are frequency, f (cycles per second) = $$ \omega_0 / (2 \pi) $$ and the period, T = $$ 1/f $$. These are not so important, but often people will specify the frequency or the period instead of the angular frequency, so they can be helpful.

Now, to get the velocity, differentiate the position, and to get the acceleration, differentiate the velocity. We have,

$$v = A \omega_0 sin ( \omega_0 t)$$ and $$a = -A \omega_0^2 cos( \omega_0 t) = - \omega_0^2 x$$.

Now, we have avoided saying what A is. It turns out, it depends on the problem, or the initial conditions. We can say the velocity or position of the oscillator at some t is something and then use the expression for v or a to find A. You can do the same thing with the phase if you want, but it is a little tedious and doesn't tell us much.

Notice the greatest velocity is at the equilibrium position (x = 0) of the oscillation. We can go one making such statements, but they are all extremely obvious if you simply plot out the position, velocity, and acceleration on the same graph.

Damping
An object that oscillates freely oscillates at its natural frequency. If it loses no energy, it will continue to oscillate forever. Damping is when an oscillating mass loses energy. There are 3 types of damping:

1) Light - The amplitude gradually decreases over time

2) Critical - The mass would overshoot 0 displacement

3) Heavy - The displacement decreases to 0 without any oscillation.

The cause of damping is frictional forces, e.g. Car suspension

Let's try to quantify this a bit. Say there is a friction force which is proportional to the velocity (this is a pretty good approximation in many cases) with constant of proportionality c. Then, by Newton's second law,

$$m \frac{d^2 x}{dt^2} + c \frac{dx}{dt} + kx = 0$$.

This equation is a little trickier to solve than without the friction. I am going to use a very nice trick which you will find throughout physics, and whenever you have similar equations. Notice that if x is a solution and y is a solution, then ax + by is also a solution, where a and b are constants (real or complex). This property means the equation is called "linear." We know that $$e^{ix} = cos(x) + i sin(x)$$. Assume x is $$Ae^{i \omega t}$$. Then we just take the real part of x and we get our answer because the equation is linear, but exponentials are so much easier to work with than sines and cosines. The equation of motion becomes

$$(-m \omega^2 + i \omega c + k) e^{i \omega t} = 0$$

So,

$$(m \omega^2 - i \omega c - k) = 0$$ or $$\omega_0 = ic/2m \plusmn \sqrt{(c/m)^2 /4 - k/m}$$.

Defining $$ \gamma = c/m$$, and remembering $$\omega_0^2 = k/m$$

$$\omega = i \gamma /2 \plusmn \sqrt{-\gamma ^2 /4 + \omega_0^2}$$.

Defining $$ \omega_\gamma = \sqrt{-\gamma ^2 /4 + \omega_0^2} $$, we have the general solution

$$ Ae^{-\gamma/2 + i\omega_\gamma} + Be^{-\gamma/2 - i\omega_\gamma} = e^{-\gamma/2} (A e^{i\omega_\gamma} + B e^{-i\omega_\gamma} ) $$.

All we do is take the real part of this with Euler's identity, and we have,

$$ C e^{-\gamma/2} cos( \omega_\gamma t + \phi) $$,

where C and $$phi$$ are just A and B written a different way. You can find them if you want, but they won't be very helpful. Notice that the oscillator oscillates with ever decreasing amplitude, but not at its "natural" frequency, but at a different frequency.

It is conceivable that $$\omega_\gamma$$ is imaginary, in which case, the entire solution is just a negative exponential! This is called critical damping, when it just turns into being an exponential instead of oscillitory motion.

Resonance
A mass resonates, when the driving frequency of oscillations is equal to the natural frequency of the object. (See Tacoma Narrows Bridge also known as "Galloping Gertie") This means that work is done to keep drive the oscillations.

If the driving frequency is less than the natural frequency, the amplitude decreases to a much smaller value.