Waves and Sound

Last week, we studied waves, sound, and music as it relates to math and physics. Two types of waves we studied are transverse and logitudinal. They differ in that the movement of the medium in a transverse wave is perpendicular to the direction of energy transport. However, the movement of the medium in a longitudinal wave is parallel to the velocity of the wave. These are both forms of mechanical waves, which require a medium to travel, as opposed to electromagnetic waves, which can travel in space due to electric and magnetic fields. Sound waves are also mechanical waves. Standing waves, or "stationary waves," occur when a wave remains in a constant position, at certain frequencies called harmonics. Music is made up of a blend of harmonics.

Standing waves occur in stringed instruments, like the guitar displayed above, that are tied down at both ends. The first harmonic of a stringed instrument is the lowest frequency at which at standing wave will occur. The second harmonic is the two times the fundamental frequency, the third harmonic is three times the fundamental frequency, and so on. The first harmonic is one half of the wavelength, as opposed to woodwind instruments where the first harmonic is one fourth of the wavelength. Woodwind instruments are those where the pipes are closed at one end. These instruments only have harmonics of odd numbers such as the first harmonic, third harmonic, fifth harmonic, and so on. For both types of instruments, air molecules come together at the nodes, where there is no air movement. In between the nodes, however, there is high air movement. Sound, in fact, is the movement of air molecules that move our ear drums. Thanks to harmonics and physics, this sound appears as music!

Light and Optics

Refraction:
Light follows the "Path of Least Time" to get to a destination in the quickest way possible. This involves 2 different mediums and a "bend." This bend is otherwise known as refraction. Refraction is when light passes through a medium and changes its path. It follows the principle of Snell's Law, where n1(sin1) = n2(sin2). In Snell's Law, n1 is the index of the angle of incidence, angle 1 is the angle of incidence, n2 is the index of the angle of refraction and angle 2 is the angle of refraction itself.

Real World Connection:

The image above depicts refraction through a pencil in a container of oil. The upper half of the pencil remains in air, where the index of refraction is 1. The lower half of the pencil  is submerged in oil, where the index of refraction is 1.47. Because the pencil is moving at a faster velocity in the air, it will slow down in the oil, making the angle if refraction smaller than the angle of incidence. If we plugged these numbers into Snell's Law, we would find the angle of refraction, which would explain the "bend" in the image shown above. A "bend" is necessary in order for light to achieve the path of least time.

Magnetism

Explain how objects like the earth and metals can be permanent or temporary magnets.

When it comes to magnets, a domain is a cluster of atoms that are magnetically aligned. A permanent magnet has domains that are all lined up in the same way. The three most common magnets are iron, nickel, and cobolt. Some objects have magnetic material and the potential to become magetic, however, are not always magnetic. They have domains, but they are not always lined up. Other objects are made of non-magnetic material. They have domains but these domains can never be lined up. In these types of objects, there are magnetic fields but no magnetic domains. In conclusion, all objects have domains. If these domains are lined up, the object is magnetic or has potential to become magnetic. If the domains are not lined up, the object cannot become magnetic.

iPad Battery Connection

http://www.mahalo.com/ipad-battery/

     The article talked about the voltage of batteries when they are fully charged. Because of last week’s lesson, these terms are much more familiar. The charge of an atom is generally neutral. When an atom looses electrons, it becomes positive. However, when it gains electrons it becomes negative

Fields of charged objects make up voltage. Voltage is the separation of charges. The lithium poylmer batteries are used because they separate the charges for a long period of time.

Voltage and charge combined make up electric potential energy. However, voltage alone is supplied by a battery. In our lemon battery lab, the lemon supplied a voltage. With a current, resistor, and voltage, a simple circuit is made and electric potential energy can be transferred.

Projectile Motion

For this week's lab, we went down to the gym and played some basketball while measuring the projectile motion of the basketball. We used the Video Physics app to create motion maps from the video of the shot at the basket. Then, we created graphs for the x and y components, the position over time and a velocity over time graph in the horizontal direction (x) and the vertical direction (y). 

The graph to the left on top shows the x position over time.  The change in position over the change in time represents the slope, which is is constant.  The change in position over the change in time is also velocity, which is constant. The graph below to the left shows the x velocity over time.  The change in velocity over the change in time is acceleration.  Because the slope of the line in this graph is 0,  there is no acceleration and no net force.

The top graph to the right shows the y position over time. The change in position over the change in time is the slope and velocity. The slope is not constant--therefore, the object is accelerating. The bottom graph to the right shows the y velocity over time. The decreasing line crosses the x-axis, and at that moment the object has a velocity of 0, which means it is at its peak. The rest of the time it is decreasing.

Forces in 2D and Circular Function

In our first lab of the second semester, we used the kick discs to test the nature of normal, gravitational, and tension forces. We measured the x and y values and used SOH CAH TOA to find the magnitude. These x-components and y-components are independent from one another. By adding up the x-components and the y-components, we can calculate the net force to analyze these forces in 2D. By spinning the disc, we created a centripetal force.  For an object to move in a circle, you need to apply one force toward the center--this motion causes acceleration. To orbit means to travel in a circular path. Satellites and even other planets orbit the earth through the centripetal force of gravity.

Newton's Laws of Motion

Lab #1: Hover Disc:

In the Hover Disc Lab, we went down to the gym foyet and discovered what gives rise to a change in motion. We observed the disco at rest, in motion, and while it was being pushed. Then, we took note of the change in motion and documented it in an interaction diagram and a free body diagram.

When the disc was at rest and in motion, there was a gravitational and normal force between each person and the earth and the disc and the earth. The normal force was going upward and the gravitational force downward in the free body diagram. When the disc was being pushed by either person, there was a gravitational and normal force between each person and the earth and the disc and the earth. There was also a normal force between the disc and the person pushing it. In the free body diagram, there was a normal force going upward and to the right, and a gravitational force going downward.

Newton's 3rd Law of Motion:

This lab allows us to act upon Newton's 3rd Law of Motion, which states that when two objects interact, they exert equal and opposite force on each other.These objects are equal magnitude, opposite in direction, and the same type of force.

Lab #2: Fan Cart:

In our Fan Cart Lab, we discovered the relationship between mass, force, and acceleration. First, we found our slope, or constant force of .15 N. Then, we added mass to our fan cart to observe the affect on the acceleration. Below is an image of our data:

Newton's 1st and 2nd Laws of Motion:

From our data, we derived the equation: force= mass x acceleration. This equation happens to be Newton's 2nd Law of Motion. This affects Newton's 1st Law of Motion, which states that an object at rest or traveling at a constant speed will continue to do so, unless a net force acts on it. 

Real World Connection:

Below is a video, which describes Newton's three laws of motion, and involves connections to our everyday life:  http://www.youtube.com/watch?v=UVdqxYyFRKY

Impulse Lab

In this week's lab, we performed a collision to determine the relationship between force, distance, and time. First, we attached our force probe and zeroed it. Then, we performed a collision between the red cart and the aluminum ring and found the momentum and measured the velocity before and after the collision. We found the velocity before to be 0.2625 m/s and after to be -0.2997 m/s. We measured our force as -.2100 N. Below is an image of our data in LoggerPro:

Through our data and class discussion, we came to the conclusion that impulse is the area of a force vs. time graph. In a collision, the momentum changes because of the impulse, measured in NxS (or force x time). No matter the mass, there is always an equal and opposite force--in any collision. When you increase time, force is always decreased. We found the equation for impulse to be J= Pf - Pi, or impulse= final momentum - initial momentum. Below is an image of our white board:

Real World Connection:

In the real world, the physics behind bowling relate to momentum and impulse. When the bowling ball comes in contact with the pins, a collision takes place. The momentum of the ball, however, changes because of the impulse. The article below further explains this theory:
http://www.topendsports.com/sport/tenpin/physics.htm

Collisions Lab

In this week's lab, we studied the difference between energy lost in an Elastic Collison vs. an Inelastic Collison. To do so, we set up two carts so that their spring launches were facing one another, so ensure that they would bounce off each other for an elastic collision.  Our carts weighed .25 kg. We measured the speed of the red cart to be .472 m/s and the blue cart to be 0 m/s before the collison. After the collision, the red cart had a velocity of .323 m/s and the blue cart to be .243 m/s. Then, we set up the carts so that their velcro sides were facing one another so that they would stick together, for an inelastic collision. We recorded the red cart to have a speed of .667 m/s and the blue cart of 0 m/s before the collision.
After the collision, the red cart had a velocity of
.031m/s and the blue cart of .431 m/s. A table with
our data is shown below:

After our experiment, we calculated the amount of energy for the elastic and inelastic collisions by finding the percent difference, which is the total energy before x the total energy after / the average of total energy before and after x 100.  We calculated the percent difference of energy in the elastic collision to be 19.608 %. The momentum in the elastic collison was 1.709 %. For the inelastic collision, the energy had a percent difference of 94.737 %, and the momentum of 16.181 %. For both types of collisions, momentum had a lower percent difference, so momentum is more conserved and momentum can help us analyze collisions.

Real World Connection: 

Above is an image of a collision in real life and this article desribes the relationship of energy and momentum that causes it. http://toppers-club.com/iit-physics/be-careful-in-applying-law-of-conservation-of-energy/

Rubber Band Cart Launcher

     In this week's lab, we were asked to determine the relationship between energy and velocity. To do so, we detained the velocity of our red glider cart, using the electronic force probe. We stretched the rubber band to a distance of .01 m, .02 m, .03 m, .04 m, and .05 m. According to the electronic force probe, we documented the velocity of the cart at each distance. We did two trials for each distance and then found the average velocity, the average velocity squared, and we noted the energy measured in joules from our experiment last week. We used our graphical analysis app to determine the graph of our data, using the average velocity squared as the x-axis and the energy as the y-axis. Our graph is shown in the image below:

Using two points on our graph, we found the slope to be 1/2. We converted our y= mx+ b formula into Energy= 1/2 mass x velocity squared.  With K as the elastic constant and Us as the potential energy, we were able to determine that Us = 1/2 k x squared.

We determined that energy in a system always stays the same and that energy and velocity are directly related, and if energy increases velocity must increase. If velocity increases, so will energy.

Real World Connection:

http://ffden-2.phys.uaf.edu/211_fall2002.web.dir/shawna_sastamoinen/Velocity&Kinetic.htm

A rollar coaster can demonstrate the relationship between velocity and energy because if the mass is constant, and the velocity is increased, the kinetic energy must also increase. The article linked above clearly demonstrates this principle as well.

Rubber Band Lab

In this week's lab, we figured out if the force it takes to stretch a rubber band depends on the amount by which you stretch it. To solve this, we stretched a rubber band to different lengths of .01cm, .02 cm, .03 cm, .04 cm, and .05 cm with the electronic force probe. With one rubber band loop, we measured the following:

.01 m = .4 N

.02 m = 1 N

.03 m = 1.9 N

.04 m = 2.7 N

.05 m  = 3.2 N

We repeated the process with a double rubber band loop and got the following data:

.01 m = 1.6 N

.02 m = 2.2 N

.03 m = 5.5 N

.04 m = 5.6 N

.05 m  = 6 N

We notice that when we stretched the band farther, the force increased. Then, we graphed our data using a best fit line. We were given Fs as the force needed to stretch the band, K as the elastic constant, and X as the distance pulled. We discovered the equation Fs=KX, also known as Hooke's Law. Our graph is shown below:

We measured our energy, area, in the shape of a triangle. Area was represented by Us, the elastic potential energy. Using the standard equation of the triangle, our equation became Us = 1/2 ( base x height). With X as the base, Fs as the height, and using K as the constant, our equation turned into Us = 1/2 (K) (Xsquared). We discovered that distance and force are directly proportional, when one increases so does the other. Here is a picture of our white board:

Real World Connection:

An example of the effects of force and distance is a slingshot. The more force used to pull back, the greater distance it will reach. Therefore, the g reater distance you would like to reach, the more force needed as well. This use of force and distance is a great example of elastic potential energy. Below is an image of a slingshot you could use: