1. Unit 1.1 Vibrations

2. A repeating back-and-forth motion about an equilibrium (rest) position is a vibration. Whenever an object vibrates so that there is a restoring force that is proportional to the displacement of the object, the vibration is called simple harmonic motion (SHM).
Examples of SHM are a pendulum swinging back-and-forth and a mass on a spring bobbing up and down.
A long pendulum takes more time to swing one
time back and forth than a short pendulum.
A spring with a larger mass on it will take more
time to swing once back and forth than a spring with less
mass.
The time it takes for one complete back-and-forth motion is called the period of the system (pendulum or mass/spring) so longer pendulums have a longer period.
The number of back-and-forth motions completed in a certain amount of time is called the frequency of the system (in other words, it measures how frequently the system vibrates).

3. If the certain amount of time is seconds then the units are called hertz (Hz) – for example 10 vibrations per second = 10 Hz.
Notice that when the period gets longer the frequency gets smaller. The period and frequency have an inverse relationship – as one increases the other decreases.
Let’s say a pendulum has a period of ½ of a second. Since it takes ½ of a second to complete one vibration it will complete 2 vibrations in one second. So the period is ½ second and the frequency is 2 Hz. This reciprocal relationship between the period (T) and frequency (f) is summarized as T = 1/f or f = 1/T
(see the table below)
Period (s) frequency (Hz)

4. As stated earlier, whenever a system (like a pendulum or a mass on a spring) vibrates so that a restoring force that is a proportional to the displacement is produced, then SHM occurs.
In the picture to the left, the distance x
is the displacement of the mass and the
restoring force is how hard the spring
pulls to bring the mass back to its
equilibrium position. The larger x is
the greater the restoring force will be,
so this is an example of SHM.
For a mass on a spring, there are only
2 factors that will change the period
(and therefore frequency) of the system. One factor is the amount of mass – the larger the mass the longer the period (and smaller the frequency). The other factor is how stiff the spring is (called the spring constant). For springs with large spring constants (like shock absorbers on cars) the period is smaller (and the frequency is higher).

5. For a pendulum there are also only 2 factors that can change the period (and frequency) of the pendulum.
One factor is the length of the pendulum. The longer the pendulum the longer the period (and smaller the frequency).
The other factor is harder to change because it is the acceleration of gravity. It turns out that if the acceleration of gravity is greater (i.e. stronger gravity) the period of the pendulum will be shorter (and the frequency will be higher). This means that if you place identical grandfather clocks on Earth and the moon, the clock on the moon will run slower than the one on Earth.
slower faster