Wave Interference Part 2- 2-dimensional Wave interference and PLD

Hey Everyone!

Now that we've talked about one-dimensional wave interference (standing waves) we can talk about something a little more complex: 2-dimensional wave interference.

In most situations, waves aren't interacting on a string, right? They tend to work across the surface of a two-dimensional space (think water waves). The two sources produce pulses at the same time, and for our purposes, with the same frequency and wavelength.

Remember how standing waves have nodes and antinodes? Well, this is true with 2D wave interference as well. The really cool thing about nodes or antinodes in two dimensions is that they line up! This forms a nodal line (or an antinodal one)! (see diagram below) Those weird grayish line on the diagram are nodal lines- they NEVER move!

A critical term to know when talking about 2D wave interference is path length difference. Essentially, the two waves produced by the two sources have to travel a certain distance to reach a given point. The absolute value of the difference between the distances traveled by both waves from the center to that point is the path length difference.

The highest possible value for the path difference is equal to the distance between the two sources (the source separation).  The lowest possible value is at any point that is equidistant from both sources (and is therefore equal to 0 wavelengths)

The formulas for Path length difference are:

Path length difference = |pathlength1 - pathlength2|

Path length = n * wavelength

Whoa, hold it. What the heck is n? Where did THAT random variable come from? Essentially, n is the path length difference in terms of wavelength.

When n is a whole number, an antinodal line exists. When n ends in ____.5, a nodal line passes through that point.

See diagram or "di a" ... never mind. You see what I did there? Red lines are antinodal, blue are nodal, and green (should be red) is the line of source separation.

The last thing I'm going to talk about before I let you amazingly patient people go to bed are the EQUATIONS we can use.

path length difference = n* wavelength


v = f * wavelength

Thanks for your time and patience, everyone! Remember to share this page and help me get more followers- so we can help more people!
Thanks, and as always,  contact me on Facebook or at Satya.root.beer@gmail.com if you want physics help!

Wave interference Part 1- Standing Waves

Hey Everyone! 

I know its been a while, so thanks for your patience!

Right... we know what waves are, how they behave, etcetcetc. Now, we can start to understand what the heck happens when two waves interact. This post is going to cover very simple, one-dimensional wave interference, and the next post (in about an hour at most) will cover more complex two-dimensional wave interference

One-dimensional wave interference between two continuous waves of an equal amplitude, wavelength, and frequency produces a standing wave. A standing wave has NO net transfer of energy from one point to another. This means that a standing wave... well, stands. 

When the crest of one wave meets the crest of the other (or a trough meets a trough), it results in constructive interference- the amplitude of the resultant wave is 2x greater than the amplitude of either of the two waves. When a crest meets a trough, however, it results in destructive interference- the amplitude of the resultant wave is less than either of the two waves. In the case of a standing wave, this simply means that the two 

waves cancel out.

There are two very important points on a standing wave: Nodes and antinodes. An antinode is a point where there is ALWAYS constructive interference, while a node is a point where there is ALWAYS destructive interference. At a node, both waves always cancel each other out- the nodes never move!

The equation we need for standing waves are: 

Where n is the number of nodes along the length of the medium, lambda is wavelength, and l is the length of the string/medium. Then, of course, there is the standard equation for waves:

wave speed = wavelength * frequency

Wavy stuff

Hey Everyone!

Nah... This isn't the kind of wave I'm talking about (unfortunately).

Today, I'm going to be talking about the Physicsy kind of waves- what a wave IS, the two types of waves, the 5 main properties of waves, and electromagnetic waves.

First off, what the heck IS a wave?

Essentially, a wave is ENERGY. Nothing more, nothing less. A wave is energy that is transmitted from one point to another through a medium- i.e. THROUGH something. That medium can be practically anything, from air (say "heloooooooooooooooooooo") to water (glugglugglug) to an electromagnetic field (more on that later on).

Every wave forces particles to move- a wave transfers energy to particles in its way. Waves can be classified to two ways based on HOW they force particles to move. A wave is either transverse or longitudinal based on how they force particles to move.


In a longitudinal wave, the energy forces particles to move in the direction of the wave's travel. Longitudinal waves cause compressions and rarefactions- basically, the medium has a higher density in a compression and a lower density in a rarefaction. The diagram below represents the density of the medium at any given point.

In a transverse wave, the energy forces particles to move up and down- this is perpendicular to the direction the wave is traveling. Transverse waves are made up of a series of crests and troughs (See below)

Electromagnetic waves are a very special kind of transverse wave. Most waves only force matter to move in two dimensions (i.e. up and down). An electromagnetic wave, however, forces matter to move in three dimensions. How does it do that? An electromagnetic wave actually consists of two waves (an electrostatic wave and a magnetic wave) that are "in phase" and perpendicular to each other (see diagram).

Whoa, hang on! How can BOTH the waves be perpendicular to the wave's direction of travel?

The answer: The waves travel in 3-dimensional space. Imagine a three-dimensional coordinate plane (with an x, y, and z-axis). x is length, y is height, and z is depth. Our electromagnetic wave is traveling along the x-axis.

The relation between any two of the three axes is that they are perpendicular to each other. For example, the x-axis is always perpendicular to the y-axis.

So, if a wave is traveling along the x-axis, its direction of travel is PERPENDICULAR to the y-axis AND the z-axis. This is how electromagnetic waves exist- both parts of the wave are perpendicular to the direction of the wave's travel, but along different axes.

Waves have 5 essential properties- every wave does all five of these things. I'm going to go over the simple ones really quickly so I can get to the more complex stuff.

1. Reflection: Basically, when a wave hits an obstacle, part of the wave's energy is reflected back. Best examples: Mirrors and echoes. If you're standing at the edge of the Grand Canyon and yell down, you hear an echo. When the sound wave hits the rock at the bottom of the canyon, part of the soundwave is reflected back to you.

2. Transmission: When a wave hits an obstacle, only PART of the wave's energy is reflected back. The rest of the wave's energy is transferred to the new medium (aka the obstacle). Example: If you're under water, you can still hear someone talking above the surface of the water.

3. Refraction: When a wave transmits part of its energy to a new medium, the wave changes directions. Example: Have you ever stuck a pencil/straw in a glass of water? The pencil looks like it bends at the water's surface.

4. Diffraction: When waves meet an obstacle, they move around it. Plain and simple. Does this seem to conflict with points 1&2 (reflection and transmission)? Why would a wave reflect off an object or pass through it if the wave could just go around? The answer: Diffraction DOES occur for large objects, but not to a very large extent. Diffraction is only noticeable with relatively SMALL obstacles. Example: You can hear a violinist play even if said violinist closes the door.

5. Superposition: When two waves traveling through the same medium collide, they affect (interfere with) each other- but only while the two waves are in contact with each other. Once the two waves separate, they have no further effect on each other. There are two main types of superposition: Constructive and  Destructive interference (saved for next post :) )

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Thanks!

Circular Motion Review

Hey Everyone!


So... tomorrow is the Circular motion unit test for IB SL Physics at Interlake. If you go to Interlake HS and don't know this- YIKES.

Today, I'm not going to write about physics concepts- I'm just going to post a review question that I made. Try doing this question AFTER you finish Mr. T's review packet. If you were able to finish everything on the review packet and you're able to do this problem, you're ready for the test.

Here goes:

A powerful motor is directly attached to a 0.2 m-diameter wheel  as shown and connected to a battery which lets it run at its full rotational speed of 5500 revolutions per minute (rpm). See diagram below. Friction is negligible.


Given this information, calculate:
a) The angular frequency of the wheel (in revolutions/sec)
b) The angular frequency of the ball (in radians/sec)

c) A piece of gum is now stuck onto the edge of the wheel. Calculate the linear velocity of the piece of gum if the motor is still spinning at its full rotational speed.
d) Find the centripetal acceleration of the piece of gum.

This system is now connected to a circular ramp, and is used to shoot balls with a mass of 0.5 kg. Given this information, calculate:

e) The velocity of the ball when shot (assuming that the motor returns to max velocity by the time the ball is shot out)
f) The time the ball is in the air
g) The horizontal displacement of the ball
h) the maximum vertical height reached by the ball

If you want to check your answers or need help with another problem, send me a facebook message, post in the Thompson's IB Physics group, or send me an e-mail at satya.root.beer@gmail.com!

Gravity

Hey everyone!

Thanks to the first few people who've joined this blog!

Today, I thought that we could spend some time talking about the force of gravity and gravitational fields, because that's a pretty important topic that we weren't able to cover in class.

Gravity is the weakest intermolecular force... it really is ADORABLE compared to all the others.

What the heck IS a gravitational field?


Simply put, a gravitational field is the force a body applies on another body per unit mass of the second body. Every object generates a gravitational field around itself- this field has an INFINITE area. Basically, every object in the universe applies a force on every other object in the universe.

So why can't I fly to Pluto right now? Why don't I feel a force pulling me towards people in the halls at Interlake? Why don't my empty soda cans throw themselves into the trash can?

The force that two objects apply on each other is dependent on two factors:
1) Mass: Basically, the larger two objects are, the greater the force they will apply on each other. The mass of  an object and the force that it applies on another object are directly proportional. Relatively speaking, my soda can has very little mass, especially compared to a really heavy object like a star! This means that the soda can and the Sun don't apply very much force on each other. If I somehow managed to get a soda can as massive as the moon, the Sun would have to apply a much greater force on it to keep it in orbit.

2) Distance: The best way to explain this idea is with magnets. The further two magnets are from each other, the easier it is to hold them apart, right? So you can say that the magnets apply less force on each other the farther apart they are. The same concept holds true for gravitational force between two objects Vs. the distance between them. Gravitational force is inversely proportional to the distance^2 between two objects.   That's why Mercury travels so much faster around the Sun than Pluto- Mercury is much closer to the Sun, so the Sun applies a greater force on it. This means that it has a greater velocity as it orbits the Sun.

How can all of these ideas be used to make a single general equation?

What this means is that the force between two bodies is equal to the product of the masses of the objects * the universal gravitational constant / (the distance between them^2).

As always, if you have a question or comment, post here or send me a facebook message!