# Simple Harmonic Motion

APPLIED MATHS SYLLABUS PHYSICS SYLLABUS S.H.M.

Simple harmonic motion of a particle in a straight line.

A really nice easy explanation or introduction to SHM

http://www.acoustics.salford.ac.uk/feschools/waves/shm.htm

So SHM is about Oscillations, kind of like waves.

The wavelength is the same length as one oscillation of a simple harmonic motion wave.

SHM takes place when a particle moves

so that its acceleration is proportional to its displacement from a point

o - but in the opposite direction.

The time it takes for one oscillation to pass is known as the Period of the SHM.

And given a Capital T, for time, to denote it.

This is a basic type of Pendulum that experiences SHM

http://www.walter-fendt.de/ph14e/springpendulum.htm

When you watch this

1. Take note of the Period
2. Watch the speed of the particle as it approaches the ends
3. Watch the speed of the particle as it approaches the mid-point
4. Press pause and work through the various values of the particle.

the displacement of 1 cycle = 4A, from center to extreme (1A), center to other extreme (1A)

http://www.walter-fendt.de/ph14e/springpendulum.htm

Some resonace through force oscillations

More Springs

http://id.mind.net/~zona/mstm/physics/mechanics/energy/massOnASpring/massOnASpring.html

Hooke’s law: restoring force ∝ displacement.

Do you remember your Hookes Law Experiment from Junior Certificate ?

What was the outcome?

The Force applied to a spring (or other elastic material) caused an extension in the length of the spring.

This extension was proportional to the size of the Force applies

Taking the extension length to be s we can say the following

F = – ks

Which we can also write in terms of mass and acceleration

ma = – ks

k is just a constant so ..

as the mass cannot change,

the acceleration is proportional to the displacement

so a is proportional to the size of the displacement, any constant will do.

We just choose omega squared ... it has no reason ... it is just a name of a constant.

These Formulae are the ones you need.

where it started from the center

when it starts from an edge E = 0

Systems that obey Hooke’s law

(Application of simple harmonic motion to include the simple pendulum.)

Watch this your own time.

Pendulum

Here's an excellent site with lots on pendulums here

http://www.sciencebyjones.com/pendulum_swing.htm

More from Walter on the Pendulum

e.g. simple pendulum, execute simple harmonic motion:

Demonstration of SHM,

e.g. swinging pendulum or oscillating magnet.

Appropriate calculations.

Title

7. Investigation of relationship between period and length for a simple pendulum and hence calculation of g.

Apparatus

Cork, Knife, String, Weight, stopwatch, Retort Stand

Method

1. Split cork in two smooth edged pieces
2. Clamp the string between the 2 edges of the cork
3. Ready the stopwatch
4. Pull the string to one side (make sure the angle is less than 5o)
5. Start stopwatch as you release the pendulum
6. Time the pendulum for a number of swings, (30 or 50 swings)
7. Find the period, and the length of the string
8. Change the length of the string and repeat the experiment

Safety Concerns

Take care splitting the cork, always cut away from the body using a sharp blade

Results / Observations

Measure the length of the string from the fulcrum, the turning point to the cog of the bob (usually the middle of the weight)

Allow the pendulum to swing for a number of full oscillations (over and back)

Maths

This is the formula, so from the data plot t2 vs length

but usually they will give you the time for a number of swings (by taking the time for a number of swings will increase accuracy of the time measurement). Therefore you must divide by the number of swings to get t (the duration of 1 swing).

To help you it is best to draw a table

then draw a graph, plotting length (m) y vs t2 x

Now find the slope (using 2 found (not given) points, not including the origin.

The slope will be the equivlent of l/t2.

Conclusions