Superposition

The principle of superposition may be applied to waves whenever two (or more) waves travelling through the same medium at the same time.

When 2 waves are in phase phase with each other (the peaks and troughs coincide with each other the effect for the resultant wave is additive.  The example on the left shows the 2 waves which oscillate at 2 units  form the central point at the same time.  When they are superposed, the resultant line is 4 units from the central point.

When the waves are out of phase, the lines above and below cancel out.  This is because the blue wave is at +2 position (above the line) at the peak and the corresponding brown wave is at the -2 position (below the line)





Basically the resultant wave is a bit like a vector sum using the distance from the central point from which it oscillates.  The first time that the resultant red wave crosses the horizontal line marked x, the 2 component waves (blue and grey) can be seen to be equal distances above and below the line.

In some cases more complex wave forms can be produced by 2 component waves.
















Most sounds are not simple.  They are often mixtures of 2 or more "pure sounds" which provide make tones.  2 tones combined will produce a beat.  This is where there is a noticeable change in volume which will make a waahaaw- waahaaw type of sound.  Beat frequency is calculated by:

A beat wave pattern is shown below.



For some use of beats in phone tones try this page.


Many musical instruments produce complex sounds rather than simple pure note.  As a result the wave form is quite sophisticated

Properties of sound waves and the wave equation.

Sound waves are mechanical, that is they require a medium to travel in.  The particles that make up the medium vibrate backwards and forwards in the same direction as the sound is travelling. Keep an eye on the red dots if you don't believe me.  If you are skillful try it with a black dot next.

As shown by the animation above, the movement of particle causes area of high and low pressure.  These are known as compression and rarefaction respectively.

The distance between 2 adjacent compressions, 2 adjacent rarefactions, or a single compression and rarefaction combined is a wavelength of a sound wave.

When represented as a diagram, the compressions are drawn above the horizontal axis and the rarefactions below.

When the amplitude increases on a sound wave diagram the volume increases. As the diagram indicates it gets louder.

In the case of sound, it travels at a fixed value in air (340 m/s).  Thus if you decrease the frequency, the wavelength will increase and the pitch will decrease (more bass).  If you were to reverse this, and increase the frequency you would find that the pitch gets higher (more treble).

Some definitions for waves

Wavelength - The distance between 2 crests, or 2 troughs, or, one complete crest and one complete trough. This can shown graphically by plotting the displacement on the vertical axis of graph and the distance on the horizontal axis.

Amplitude - The strength of the signal produced by a wave.  It is represented by the displacement from the horizontal axis.  The greater the amplitude, the stronger the signal, the brighter the light or the louder the sound.

Crest - the part of the wave at the highest point from central point that it will oscillate around.

Trough - the part of the wave at the highest point from central point that it will oscillate around

Period - the time it takes one wave to pass a set point.  Note that if you graph it now, the horizontal axis is now time, not length.  Period is denoted by the symbol T.

Frequency - the number of waves that will pass a set point in one second.  The units to describe this is Hertz (cycles per second).  If you can imagine all the waves in the figure above are travelling at the same speed, you can see the idea with frequency. Frequency is denoted by the symbol f.  

There is a mathematical relationship between frequency and period.  It is as follows:

    and 

What are waves?

So what are waves?

There are a few types of wave that we will study in this unit.  However they all share one common feature.

A wave provides a means to transfer energy from one source to another without the physical movement of particles from one place to another.

Waves can be broken into 2 broad categories:

Mechanical waves – where the energy travels through a physical medium. Examples include water ripples and waves, sound and those that travel through ropes, springs etc.

Non-mechanical or EM waves.  These waves travel by disturbance on electrical and magnetic fields and so can travel through a vacuum.

Longitudinal wave particles vibrate backwards and forwards along the same axis which the wave travels.

In the case of transverse waves the vibration occurs  at right angles to the direction of wave travel 

In order to show features of waves of all types in a form that conveys relevant information we use a sine wave.

Sine waves

The simplest possible wave is called a sine wave. (The shape of a graph of the function y = sin x)

Car Safety

There are a number of safety devices that can be used to reduce the chances of a serious injury in a car when a collision occurs.  These can also be related quite easily to Newtons Laws.

1. Air bags (Newtons 2nd Law describing momentum).

While many of us thing that Newtons 2nd Law is just about F=ma, Newton in his original description stated 

An applied force is equal to the rate of change of momentum.

Recall that impulse is the amount of force applied over a given time.  

This will change the momentum of the object.  When a collision occurs, a change in momentum is inevitable. But it is dependent on the time taken.  So if we increase the time that this change in momentum takes, we can reduce the force that can affects people inside the car.  This is where airbags go to work.



Apart from reducing the applied force, airbags spread the force across a larger area meaning that rather than being concentrated at one point of impact, it will be more diffuse or spread out.  So while damage or injury may still occur, it will be less, but spread out more.


Crumple zones work on a similar principle wit regards to impulse and momentum.  They also have the added benefit if absorbing a lot of the energy in a collision and as a result, rather than an elastic collision where cars would bounce off each other, resulting whiplash injuries, they reduce this by having steering systems fold up and in this case the front end of the car distorting.




Seat belts have a lot more to do with Newton's 1st Law dealing with inertia.  While a person is in the car they are not part of the car.  Thus when the car stops, the person will want to keep moving.  In the case of sudden stops the effect becomes more pronounced.  Seat belts made of wide webbing again will apply a force on you to stop you from flying out of the seat in a collision, but they do it over a larger area and so reduce the chances of serious injury.  The spreading of force can be enhanced by widening the seat belt even further as pictured on the right here.




Unsecured objects can be seen to have very dangerous effects as shown by the picture below.  Inertia becomes more problematic as the mass increases. Even relatively small items such as torches, pieces of sporting equipment or shopping are potentially lethal in a collision because the will continue moving.  One blow to the head could be debilitating or even fatal.

Conservation of Momentum

If we consider Newton's 3rd law, "For every action there is an equal and opposite reaction", it can also be said that forces occur in pairs.  If we consider the 2 skaters, we can see that their momentum is zero, and when they are added together, it equals zero.

The skaters both push and so according to Newtons 3rd law, equal and opposite forces will apply

Now because they exert the force on each other for the came time we can say:

which can be rewritten as:

Rearranging we get:

Which in turn leads to the conclusion:

This makes perfect sense when you remember that momentum is a vector quantity.  Therefore if one object is moving in the opposite direction relative to the other, its value will be negative.

In the scenarios presented to you, it will often involve 2 cars colliding in one form or another.  There are a number of possibilities that may happen as a result of these collisions.

1. The case lock together (in this case we treat them as a single object with mass of the 2 cars combined)
2. The cars are stationary
3. The cars may bounce off each other and move in the opposite direction.

Momentum

Momentum can be described as "mass in motion". Therefore it is proportional to  both the mass of the object and its velocity.  Put simply:

Where p is momentum, m is mass and v is velocity.  Momentum is a vector quantity and has the units Newton seconds.  Momentum can also be defined with a slight alteration to Newton's 2nd Law.

Take:

The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma

alter to:

The rate of change of momentum is directly proportional to the magnitude of the net force and the direction of the net force.

F = ma  can be rewritten as:

which can be rewritten as:

Impulse is known as change in momentum.  Impulse can also be written as below:

From this point, we will move onto conservation of momentum. 

Conservation of Energy

There are plenty of examples of the conservation of energy.  This post will simply look at the changes of gravitational potential energy and kinetic energy.

Within a closed system the amount of energy remains constant and energy is neither created nor destroyed. Energy can be converted from one form to another (potential energy can be converted to kinetic energy) but the total energy within the system remains fixed.

Let's use the example above.  The ball on a frictionless pendulum will swing forever.  However, the energy is converted repeatedly as the ball swings back and forth.  At the highest point of the swing, it stops moving for an instant. Kinetic energy is therefore zero, and because its at its highest point, the potential energy is greatest.  At the bottom of the swing when the pendulum can go no lower, the kinetic energy is greatest and the potential energy is at its lowest.  Therefore we can see that the energy has been converted.

An example that we studied in the lab was the falling object through the light gates at certain heights.  Some groups found that nearly all of the calculated gravitational potential energy had been converted to kinetic energy which was measured and calculated.

A great example of conversions of energy is a hydro electric dam.  Here we have potential energy being stored behind the wall of the dam, which is converted to kinetic energy when travelling down the penstock and the turbine and then electrical energy at the generator.  

Kinetic and Potential Energy

Kinetic energy refers to the amount of energy a moving object has.  The greater the mass and the greater the kinetic energy. 

The formula for kinetic energy is:

The units of energy are Joules (J).  It is a scalar, not a vector quantity.

This also means that the faster an object travels, the amount of energy will increase exponentially. This is one of the many reasons that we have low speed zones on certain areas such as school.

Similarly, the more massive an object is, the more kinetic energy it has if it is travelling at the same velocity as a smaller one. 



Another concept you need to be aware of is the idea of work.  Work is done when a force is applied to move an object a certain distance.  So for our friend below , he has to overcome friction to get the box to move.  He will also have to push it the entire distance it has to move.  The formula for work is:

Not only does work involve moving objects, but also with moving ones.  You should be aware that a force will cause an object to undergo acceleration.

As a result, the force applied over a certain distance will result in a change in kinetic energy.  Thus:

where u = the initial velocity and v = the final velocity.

Another way in which work can be done is from lifting an object off the ground.  The work done is turned into gravitational potential energy.  Recall that F=mg for falling objects and displacement can also mean height.  Therefore:

 The implication is that the gravitational potential energy will be converted to kinetic energy is a lifted object is allowed to fall.  This will be the subject of the next post.

Newtons 3rd Law

Lets look at a scenario where a car is towing a caravan.  The car weighs 1500 kg and the caravan weighs 1000 kg.  If the car exerts a force of 7500 N, we would expect according to Newton's 2nd Law (F=ma) that the 2 objects will move off at 3 m/s/s.

However, only the car is providing the force for the forward motion, not the van.  Since the car only weighs 1500 kg, why doesn't it move off at 5 m/s/s ?

This is where Newton's 3rd law comes into play.  Simply stated, it is:

For every action there is an equal on opposite reaction.

Firstly we look at the combined system.  Using Newtons 2nd Law we calculate this correctly.

BUT.....

1. When we look at the 2 parts of this system individually, we can say that the  car has a force of 7500 N exerted by the engine to the right which will drive it forward.
2. If the caravan is accelerating at 3 m/s/s, it must have a force of 3000 N being exerted on that.

If we then apply Newtons 3rd law, the caravan must be exerting a force equal and opposite to the car (that is 3000 N from right to left, equal and opposite to the direction of the motion), the net force is now

7500 N - 3000 N = 4500 N

So when we apply this quantity of force with the mass of the car, you can now see that the acceleration of the car will be 3 m/s/s.

Just goes to show how brilliant Newton was.





Newton's 3rd law applies not only to pulling but pushing forces as well.  So if we were pushing a red Ferrari with our own red Ferrari, the car being pushed would exert and equal and opposite force on the car doing the pushing.  When something is suspended the rope etc exerts an equal and upward force on the object hanging from it.

We have all probably experienced Newton's 3rd law when we have pulled on a rope, pushed a bike or something else you have ridden or even pushed on the wall.  The classic example is when a person pushes on a wall while standing on a skateboard.

Commonsense will tell you you will move away from the wall. But this is because as they push on the wall, the wall pushes back.  When you consider the direction they travel in relation to the force they apply, Newton's 3rd Law again makes good sense.