April 24: Lab 15 Collisions in Two Dimensions

Lab 15 

Collisions in Two Dimensions

Ricardo Gonzalez and Peter

April 19 and 24 2017

The goal of this lab is to verify if momentum is conserved in a collision of two dimensions.

1. Theory
We know when we looked at one dimensional collisions that momentum was always conserved. We want to somehow show that the momentum of a collision in two dimensions is also conserved. Using our knowledge of one dimensional systems, we will apply these laws to a two dimensional collision.

2. Procedure
We first began by selecting three marbles, two of the same mass and the third of a smaller mass. We had our apparatus set up by the lab tech Maria. The apparatus consisted of a flat-stationary glass table. After leveling the table, we used our phone to capture a collision using the same mass marbles. With one marble stationary, we rolled the second marble to collide with the stationary marble. We captured the collision with our phones that captured 240 fps (frames per second). We then did the same procedure for rolling the larger mass but this time to collide with a stationary smaller mass, also recorded with the same 240 fps.
With the logger pro software open, we inserted the video capture of the first collision of equal mass marbles, we traced the position of the marbles in first the x-direction and then in the y-direction. The same tracing in the x- direction and y-direction was done on a separate file for the second collision with the larger mass and smaller mass marbles.

3. Measured Data
Mass of the larger pass marbles = 0.029 kg
Mass of the larger pass marbles = 0.007 kg 

Initial Momentum = Final Momentum so therefore

1) Initial Momentum(x-direction) = Final Momentum (x-direction) and
2) Initial Momentum (y-direction) = Final Momentum (y-direction)
 Where momentum is = mass×velocity

4. Results 

Collision 1: Graph with the slope of the positions before and after the collisions in the y direction.

Collision 1: Graph with the slope of the positions before and after the collisions in the x-direction.

Collision 1: The point taken in respect to time. The blue dots represent the trajectory of the rolled larger marble. The red dots represents the trajectory of the second marble.

Collision 2: Graph with the slope of the positions before and after the collisions in the y-direction.

Collision 2: Graph with the slope of the positions before and after the collisions in the x-direction.

Collision 2: The points taken in respect to time. The blue dots represent the trajectory of the rolled larger-mass marble. The red dots represents the trajectory of the second smaller-mass marble.

5. Analysis
In the photos above, each box shows the slope of the line, the velocity of that section. Notice we have 4 slopes in each photo. This is because in each direction, the x-direction and y-direction, we can find the momentum of both marbles before and after the collision. By using the velocity of the selected mass in each direction both before the collision and after the collision, we can find the momentum. As we see in the table above, in the momentum was completely conserved in the x-direction but was not exactly equal in the y direction. During the second collisions with two different mass marbles, we see that the final momentum in the x-direction is not exactly true. The difference of value of our initial momentum and our final momentum is about 0.00036 kg*m/s or 0.36 kg*mm/s which we can say is a very small number. Our experiment was measured in m/s and so a change of 0.00036 kg*m/s is close enough for us to ignore. In the y-direction, our change of momentum was 0.0006 kg*m/s or 0.9 kg*mm/s which is equally a very small number in change.

6. Conclusion
Although our data suggests that momentum was not conserved, we are going to assume that momentum was in fact conserved for a one reason,, our experiment was by no means perfect. When plotting the points on the logger pro software to trace the trajectories of the marbles both before and after in each collision, we are sure that every single point was not placed exactly on the marbles trajectory. The amount of outlier points is ridiculously large when we look at the graphs the points traced out. Due to these errors in it is absolutely safe to say that our experiment was far from perfect. These errors suggest that the change in momentum due to errors was huge. Since we cannot perfectly trace out the trajectory of the center of masses of each marble, we say that momentum is conserved. 

April 19: Lab 14 Impulse-Momentum

Lab 14

Impulse Momentum 

Ricardo Gonzalez, Peter, and Hannah

April 19 2017

The goal of this lab is too

April 14: Lab 13 Magnetic Potential Energy

Lab 13 

Magnetic Potential Energy

Ricardo Gonzalez and Peter

April 17, 2017

1. Objective

We have equations for both gravitational and elastic potential energy, yet we don't have an equation for that of magnetic potential energy, so we must somehow show that the conservation of energy applies to a magnetic-system. 

2. Procedure

We  first had the apparatus set up for us by the Lab-Tech Maria. The apparatus was set up by having a glider placed on an air track. On One end of he air track, a magnet was placed and similarly a magnet was placed on the glider. The air track was leveled as much as possible.

The air track system was turned on, we lifted one side (the opposite side of he air track where the magnet is placed) at a small angle. At that angle, the glider would be a distance x between the two magnets. By increasing the angle , the x distance between the magnet on the glider and the magnet on the air track decreases. We found a total of 6 distances x with angles θ.

With the aluminum deflector on the glider (as it was on for part 1), we placed a motion sensor just behind the air track. The motion detector was connected to our laptop for the Logger Pro software. We placed the glider 0.01 meters from the magnet on the air track and the magnet on the glider. We hit collect on the logger pro software and measured a total distance x(1) as the glider was not moving with the magnets 0.01 m away from each other which helped us find the "separation" from the two magnets. We set up the motion detector so that it would record 30 measurements per second. We set a new calculated column for "separation" which would be our "position" minus "the distance behind the magnet om the air track to the motion sensor." 

The distance in cm between the two magnets

3. Measured Data

4. Calculated Data
 

The force exerted from the magnet is equal to the force of gravity pushing the cart down the air track.

 

Example calculation

F(mag) = (0.339 Kg)(9.81 M/s^2)sin(14.3°)= 0.821 N

Velocity before impact = -0.44 m/s^2
Kinetic energy before the impact = 0.5(0.339 kg)(-0.44 m/s^2)^2 = 0.033 J

Velocity after impact = 0.40 m/s^2
Kinetic energy before the impact = 0.5(0.339 kg)(-0.40 m/s^2)^2 = 0.027 J

"separation" = "position" -total distance from motion sensor to glider - distance between two magnets
"separation" = "position" - 0.373 m - 0.100 m
"separation" = "position" - 0.273 m

5. Graphs

Graph showing the F(mag) vs r

The position graph of the glider with the magnets set a distance 0.01 m apart

Graph showing the velocity of the glider.

Graph 1: When the "separation" =is set to "position" - 0.273 m

Graph 2: When the "separation" =is set to "position" - 0.272 m

Graph 3: When the "separation" =is set to "position" - 0.271 m

6. Analysis

We set up another calculated column for the Kinetic energy, Potential energy, and total energy of the system as functions of time. The potential energy calculated column used the equation we derived where R is our "separation," Lastly we made a calculated column for the total energy in the system which is the Kinetic energy + the Potential magnetic energy. In the last three photos above, we can see the KE, PE(mag) and total Energy under the same graph.

We initially were given graph 1 and as we can see the potential energy of the magnet looks like it is nearly double the value of the kinetic energy of the glider resulting in a large total energy. We do know that the potential energy of the magnet is dependent of the "separation" value. So if we can assume we were not precise when we measure our distance between the two magnets, we can change the 0.273 m to 0.272 m (27.3 cm). What we can see from graph 2, our magnetic potential energy has decreased but if we change the separation to a tenth less of a centimeter, we can see that in graph 3, our value of Total energy is much  more linear than before.

7. Conclusion
In our objective, we stated that we were not given an equation for the magnetic potential energy, however we can measure it as we have shown. Although this experiment was not perfect, we made minor adjustments in our "separation" which yielded a better fit for our total energy curve in respect to the potential energy of the magnet and the kinetic energy of the glider. In theory, energy is not conserved and as as we can see in the graphs 1-3, the total energy before the collision was larger than the total energy after the collision. One source of error in our measurements was clearly the distance between the two magnets when we had to set up the "separation" calculated column. the magnets were taped onto the glider and the air track using tape which covered the front faces of the magnets. Air resistance was ignored but probably shouldn't have been. When we gave the glider an initial velocity, the aluminum deflector may have had a surface area large enough to present a reasonable opposing force to the motion of the glider.

April 10 Lab 11: Work-Kinetic Energy Theorem

Lab 11

Work-Kinetic Energy Theorem

Ricardo Gonzalez, Peter, and Hannah

April 10, 2017

The goal of this lab is to show that that by using four different experiments, we are able to prove that the work done is equal to the change of kinetic energy. 

1. Theory

We know that the work done by an object is equal to the the force*distance

W=F*Δx

We also know that the change in Kinetic Energy is equal to one half the mass times velocity squared

KE = 0.5mv(final)^2-0.5mv(initial)^2

Through these experiments, we will prove that theoretically, the Total work is equal to the Change of Kinetic Energy.  Work(total)=ΔKE

2. Procedure

Experiment 1: Total Done by a Constant Force

To set up the apparatus, we placed a cart track system on top of the table. With one side of the track overhanging from the table, we clamped a pulley. A mass of 0.50 Kg was placed on top of the cart. With the track leveled on top of the table, by placing pieces of paper under the track until the cart did not move in any direction. The force sensor was connected to the lab Pro which was connected to our computer. The Logger Pro software was started up and the Force sensor was Zero-d when held both horizontally and vertically. Next, we placed the force sensor onto the cart. A motion sensor was placed on the other end of the track , just behind the carts direction of motion. The motion sensor was also connected to the Logger Pro software. A string was attached from the motion sensor on the cart, placed over the pulley and attached to a hanging mass of 0.05 Kg.

To execute the experiment, we will hit collect on the logger pro software just before we release the cart and it accelerates down the track.

We placed two masses in front of the cart in order to not allow the cart to start moving as we took the necessary photos and set up the software for the experiment. 

We created a calculated column on the Logger Pro software fro the Kinetic Energy and added the KE to our Force vs Time graph.

Experiment 2: To find Total Work by a Non-constant Force

Using the same apparatus, we made a few adjustments. The pulley was un-clamped and put to the side. On the table we placed a clamp and a vertical rod on the clamp. The force sensor was unattached from the cart, once again Zero-ed both horizontally and vertically, and the placed onto the vertical rod. The string was replaced by a spring that was connected to the force sensor and the cart. The mass was taken off of the cart and we taped a piece of paper onto the cart in such a way so that the cart can be noticed by the motion sensor.

To execute the experiment, we will hit collect on the logger pro software and by hand, move the cart forward stretch the spring a distance of 0.60 meters. 

Experiment 3: Kinetic Energy and the Work-Kinetic Energy Principle
We will use the same apparatus as in Experiment 2.

To execute the experiment, we will stretch the spring to a distance of 0.60 meters. By using hitting collect and not moving the cart, we were able to find the distance the cart is from the motion sensor. We will Hit collect on the logger pro software and release that cart.

Experiment 4: Work-KE Theorem
For this experiment, we used a video file that showed a professor who used a rubber band to accelerate a cart that  through two photogates at a distance of 0.015 m apart.

3. Data
Experiment 1
Mass of cart = 1.128 Kg (includes the 0.50 kg mass and Force sensor)
Hanging Mass = 0.05 Kg

Experiment 2
Mass of cart = 0.506 Kg

Experiment 3
Mass of cart = 0.509 Kg (including the mass of the paper with tape)
Distance of the cart from the motion sensor is 0.595 m

Experiment 4
Mass of cart = 4.3 kg

4. Results
Experiment 1

Integral of part of the Force vs Position graph 

Integral of a larger part of the Force vs Position graph 

Experiment 2

The Integral and slope of the Force vs Position Graph. 

The calculated value of Work on the spring. 

Experiment 3

Force vs Position graph with the KE vs Position graph

The work and Change KE at various positions. 

Experiment 4

5.  Analysis
In experiment 1, we wanted to prove that the work done by a constant force is equal to the kinetic energy at the same time. When we look at the Force vs Position graph, and we take the area of the force line of a part from the entire motion from the initial distance to say x, and we look at the value of kinetic energy at x, we will roughly get the same value. In the first photo under the results section, experiment 1, we can see that the area under the force line gives us a value of 0.1101 N*M and the KE at the same Position is 0.098 J. With a difference of 0.0121 j,or 10%, the value of our Work are approximately close to our value of Kinetic Energy. To reinforce this data, we have found the Work to a larger position and found that the Work was equal to 0.2299 N*m and the KE at the same position was 0.204 J. With a difference of 0.0259 J, with a 11% error, the values are realistically close to one another.

In experiment 2,
To find the spring constant, we took the slope of the Force vs Position graph which was 3.414 N/m. By using the spring constant and the known value of x, we were able to find the theoretical value of work. The theoretical value of work was 0.615 J. However we can also find the work by finding the area under the curve on our Force vs Position graph. The value of total work from the spring is 0.7068 J. Respectively, or theoretical value was 13% off from our real value.

In experiment 3, we took the integral of the Force over multiple positions and looked at the Change in KE at the same position. By looking at the KE in respect to the Work of the spring, we can see that the values are relatively close to one another.

In experiment 4, we can see that when we found the area under the graph, we got a value for work and likewise, by using kinematics, we were able to kind the KE of the cart at the same position. The area gave us approximately 25.3 Joules while the theoretical value of Kinetic Energy at the same position was 23.8 Joules.

6. Conclusion
In conclusion, we have proven that the Kinetic energy of a system is equal to the work done by the same system. The reason for such small deviation in our experiments are due to the work done by friction of the cart/track and that of the pulley. With all else constant, small deviation are going to be present. 

April 5: Lab 10 Work and Power

Lab 10

Work and Power

Ricardo Gonzalez

Josue Luna

April 5, 2017

The overarching goal of this experiment is to use known knowledge of work and power to measure real life situations.

1. Theory
We know that when we apply a force on a mass over a given distance, work has been applied by the source of the force. Also, when we we apply work, the power exerted is given by the amount of work divided by the time.

2.  Procedure
Part 1
We began by setting up our experimental apparatus. A long wooden board was placed on top of a second floor outside through the railing. At the end of the board, a small pulley was pre-attached, however we ran a rope through it and let it overhang to the bottom. At the bottom, we attached one end of the rope to a backpack with a designated mass (m). Using gloves, we pulled up on the rope while another student simultaneously held down the end of the board to the floor and a third student would measure the time it took to raise the backpack. The pack was raised to the floor on the second story.

Part 2

April 5: Lab 9 Centripetal Force with a Motor

Lab 9

Centripetal Force with a Motor

Ricardo Gonzalez 

Josue Luna

April 5, 2017

The overarching goal of this lab is to understand there is a relationship between the angle of a horizontally spinning apparatus and the angular speed applied to the apparatus.

1. Theory
To understand the concept of this lab much easier. When we were kids, well atleast when i was, we have had some object attached to a sting and spun over our head horizontally. When we applied a larger force, the apparent  speed of the object increased and we observed the object was getting closer to moving directly horizontally to the pivot point. Similar to what we have done as kids, we can use an apparatus to apply an ever increasing force to create a larger angular speed and likewise, we will see the pivot angle will increase.

2. Procedure
We set up an apparatus in which a tripod will hold a motor on top. The motor is attached to a vertical rod that will spin as the motor is spun. A horizontal rod is attached to a vertical rod, and a string is attached to the horizontal rod. At the other end of the string we have attached a mass of m.

To conduct the experiment, we were to find the measurements needed from the apparatus. These measurements include the radius from the horizontal distance to the attachment point of the string, the height to the horizontal rod, the height to the bottom of the mass, and the length (L) of the string to the bottom of the mass. 

3. Measured Data

Radius (R) = 0.75 m

Sting Length (L)= 1.595 m
Height of apparatus (H)= 1.795 m
height to the bottom of mass (h)=  0.20 m

In order to find the Theoretical angular speed by only utilizing the measurements found for each trial, we will use the sum of the forces in the x and y directions. What we derived is an equation that is the equal to the angular speed W. 

To find the angle that the mass creates from the pivot point can be found using the laws of cosine. 

4. Results 

Our Measurements for each trial along with the calculated results for the angle, experimental and theoretical values of omega (angular speed).

A graph that shows the relationship between omega vs Theta (Angular speed vs Angle)

5. Analysis
By analyzing our data in the table shown above, we can assume the experiment was not perfect, as we can see the experimental and theoretical values for angular speed begin to deviate from each other more and more as the power is increased to spin the object. When we look at the graph of the omega vs theta, we can see that the line created is not linear. in fact, when we took an polynomial fit to the second degree, we can safely assume that as the angular speed continues to increase, the angle will even out and stay constant. Theoretically, we know it is impossible to increase the angular speed to the point where the mass will now move any higher than the pivot point, therefore the graph shown is correct in our representation of the angular speed vs the angle created.

6. Conclusion
In conclusion, we know the apparatus is not by any means ideal. Therefore, we can safely assume there is some friction created by the system that increases as the force to increase angular speed increases causing a reasonable deviation from our experimental and theoretical value of angular speed. As far as our graph that creates a relationship between angular speed and the angle created by the system, we can see that physics has not changed. As the angular speed increases, the maximum angle created by the system becomes harder to achieve. Therefore a larger angular speed is needed for the system to be perfectly horizontal. 

March 29, 2017: Lab 8 Centripetal Accelertaion vs. Angular Acceleration

Lab 8

Centripetal Acceleration vs. Angular Acceleration

3-27-17

Ricardo Gonzalez

Josue Luna

The goal of this lab is to find a direct relationship between the angular speed and the centripetal force. Angular speed represented as (⍵) can be related to the Force Centripetal. 

1. Introduction/Theory

For our experiment, we will find a direct relationship between Angular speed and Force centripetal. To do so, we have to understand what is happening. As a surface spins with a mass placed on it, the mass experiences a few forces. As the mass is spun at a constant angular speed, the force wanting to pull a mass our of its surface is due to a Force centripetal. When we tie a string from the center of the surface to the mass and increase the angular speed, the force wanting to pull the mass out of the surface is going to be equal to the tension in the string. This is best shown by a free body diagram below. 

Free body diagram for a mass as the surface rotates with a constant angular speed. 

We know that as long as the surface is rotating only in the horizontal direction, the only forces acting on the mass is its weight due to gravity and the normal from the surface to the mass. Using Newtons Second law, the F(centripetal)=ma(centripetal). We also know that the acceleration centripetally is also equal to the radius times angular speed squared (ac=rw^2). By substituting rw^2 for the a(centripetal) in Newtons second law, we get an equation of F(centripetal)=mrw^2. Assuming the radius and mass are held constant, we are left with an equation that, in hopes, relates angular speed and centripetal force. 

2. Procedure

By using a power supply generator and a previously constructed apparatus in which a circular surface is spun using a series of wheels and belts that is powered by a power generator, we are able to simulate a rotating disk at constant angular speed. By using the logger pro software, we zero-d the forces both vertically and horizontally. On top of the surface, we tightened down the force sensor and taped a scale onto the surface. We also placed a piece of tape at the edge of the surface in such a way that by using a photogate, we would be able to measure the period of one revolution. The photo gate was connected and monitored as well on the  the logger pro software. One end of a string was tied to the force sensor and the other end was tied to the mass with a radius that would vary and is measurable using the taped scale on the surface. 

A 2-D representation of our experiment

The apparatus used of our experiment

3. Measured Data
We measured our radius in inches and then converted those measurements to meters in order for the measurements to be in SI units.
The period was measured by taking the average of 10 revolutions. 

To calculate W^2, we set the equation W^=(2*pi/T)^2 where T is our period for one revolution.
  

4. Results

A graph of the Force vs Mass

A graph of the Force vs W^2 (angular speed)

A graph of Force vs Radius

The equations solved for F/M= RW^2 and
F/R=MW^2

The equation solved for F/W^2=MR

5. Analysis
To prove the mass is held as a constant, we want to graph the Force vs mass. To do so, we then assume the angular speed and radius are held constant. By doing so, we can graph the trials that contain the same angular speed (the same power supply) and radius and varying masses. The graph as shown in the photo above, shoes a linear slope of 8.937 N/kg, and a correlation of 0.994. When we mathematically calculate the right side of the equation Force/Mass = Radius*Angular spped squared, we get 9.424 m*rad^2/s^2. It is obvious that 8.937 does not equal to 9.424. However the correlation of the results shown in the graph are very close to 1 and thus proves that as the mass increases, so does the force and so we can then say the mass can be held as a constant.

To prove the radius is constant, we want to graph the Force vs radius where the radius is changing. Similarly, the trials we will use will will be when the mass is held constant and the power supply is held constant, and thus assuming the angular speed is constant. By analyzing the graph, we can see the slope of the points is 2.649 N/m and a correlation of 0.981. When we solved the right hand side of the equation Force/Radius=Mass*Angular Speed squared, as shown in the photo above, we get an answer of 6.11 kg*rad^2/s^2. The slope of the line of the graph should be equal to the right hand side of the equation, but is once again not the case. With a correlation close to 1, we can then assume the radius can be held as a constant.

Lastly, to prove the angular speed is directly related to the Force applied when assuming the mass and radius are held constant, we graph of Force vs W^2. The trials used were those that contained the same mass and same radius from the center and the trials where the power supply changes. As we analyze the graph, we can see that the slope of the points gives us 0.128 N*s^2/rad^2 and a correlation of 0.987. When we solve the right hand side of the equation Force/Angular speed squared=Mass*Radius, we get an answer of 0.141 kg*m. For the first time we have found that the slope of the graph is equal to the right hind side of the equation, with a very minimal amount of error. This alone tells us that the Force is directly related to the angular speed squared.

6. Conclusion

In conclusion, we have found that when holding the mass and radius constant, increasing the angular speed by changing the power supply, the both sides of the equation F/W^2=MR is held true. This is because when we set the R&W^2 and M&W^2, the angular speed was not the same any given instance in either scenarios. Only when the Mass and radius were held constant did both sides of the equations very closely equal one another. The apparatus used was not by any means perfect and thus has created deviations within the periods of all trials.