# oscillations

## Simple Harmonic Motion Graphs

Here’s my attempt at animating 5 graphs for simple harmonic motion together in one page.

From left column:

$$v = \pm\omega\sqrt{x_o^2-x^2}$$

$$a = -\omega^2x$$

From right column:

$$s = x_o\sin(\omega t)$$

$$v = x_o\omega \cos(\omega t)$$

$$a = -x_o\omega^2 \sin(\omega t)$$

And here is the animated gif file for powerpoint users:

## Braking of a Magnetic Pendulum with Copper Plate

In this video, we will observe how induced eddy currents in a copper plate slow down a magnetic pendulum.

When the pendulum is set in motion, it usually oscillates for quite a while. This pendulum consists of a strong magnet.

If we slide a copper plate underneath the magnet while it is in motion, the magnet comes to a stop quickly. Note that copper is not a ferromagnetic material, which means it does not get attracted to a stationary magnet.

As the magnet moves across an area on the copper plate, the change in magnetic flux induces eddy currents on the plate. These eddy currents flow in such a way as to repel the magnet as it approaches the plate and attracts the magnet as it leaves the plate, therefore slowing the magnetic pendulum.

When we pull the copper sheet out from under a stationary magnetic pendulum, the eddy currents will flow in such a way that it becomes attracted to the copper sheet.

Moving the copper sheet to and fro at a certain frequency (the pendulum’s natural frequency), the magnetic pendulum can be made to oscillate again.

## Tuning a Guitar using Resonance

There are many ways to tune a guitar. Many musicians would have tuned a string instrument using a tuning fork at some point. However, the conventional method of tuning with a tuning fork is by listening to beats while adjusting the tension of the string. The tuning fork is of a known frequency which corresponds to a note. For instance, 440 Hz corresponds to an A-note. When the A-note string is slightly out of tune, such as having a frequency of 438 Hz, the resulting sound pattern (called beats) will have a frequency that is the difference between the two frequencies, i.e. 2 Hz. Hence, the aim of tuning by listening to beats is to adjust the tension of the string until the beats disappear.

An alternative method, which is the one we shall attempt in this demonstration, is to run the vibrating tuning fork along the E-string (this first from the top) until you reach the bridge between the 5th and 6th frets. You should expect to hear a loud resonating sound there. Otherwise, adjust the tension until you do.

All the other strings are tuned with respect to that first string.

Explanation

Resonance is the phenomenon where the frequency of the tuning fork (driving frequency) is equal to the frequency of the string (natural frequency) and maximum energy is transferred from the tuning fork to the string. The string will hence oscillate with the maximum amplitude.

## Resonating Pendulums

The purpose of this demonstration is to teach the conditions and effects of resonance.  Our setup includes three sinkers hanging from a rod. I give credit to my colleague Alan Varella for showing me this demonstration when I first started teaching.

What I do with my class is that I would jokingly announce that I can use telekinesis to cause any sinker to oscillate at will while keeping the others still. This provides some entertainment and after I do the first demonstration, I can even challenge one of them to try to do the same or ask the class for suggestions on how the phenomenon can be repeated.

Materials

1. 3 fishing sinkers or pendulum bobs,
2. Some nylon string,
3. A rod of about half a metre’s length.

Procedure

1. Tie each sinker to a piece of string of varying length and then tie the string along the rod at roughly the same distance apart.
2. By holding the rod at one end so that the three sinkers dangle in front of your hand, you can begin to move the rod slightly and slowly at first. The hand should be moving so little that it goes unnoticed.
3. Gradually increase the frequency of the slight hand movement and when you see the sinker with the longest line begin to start oscillating with larger amplitudes, stay at that frequency.
4. Once you are satisfied with the oscillation of the first sinker, you can try obtaining resonance with the other two by starting over again with a higher frequency this time.

Science Explained

Resonance occurs when the frequency that you are driving the rod with is now equal to the natural frequency of the sinker on a line. Meanwhile, the other two sinkers do not oscillate as obviously as the one with the longest line.

Resonance is the tendency of a system to oscillate at larger amplitude at some frequencies than at others. A simple example will be a child on a playground swing being pushed by her friend standing at one end of the swing. If the friend pushes the child on the swing every time the swing reaches one end, more energy is being introduced each time, causing the child to swing higher and higher. Notice that a swing will always oscillate about the same frequency, with the weight of the child making little difference. At these natural frequencies of oscillation, even small periodic driving forces can produce large amplitude oscillations.

For the case of the sinker-and-line system, the frequency f at which resonance takes place for each sinker should be given by the formula

$$f={\frac{1}{2\pi}}\sqrt{\frac{g}{L}}$$

where g is the gravitational acceleration and L is the length of the line.

Hence, the pendulum with the longest string will resonate at the lowest frequency among the three.