Friday December 5, 2014: Mass Spring Ocillations.

Purpose: Find the relationship between period and oscillation.

Apperatus: Spring, motion sensor, a mass and logger pro.

What We Did: Professor wolf informed us that 1/3 0f the mass of a spring actually goes into the mass for a period of oscillation. We were part of group 1 where we all had to get our mass to equal 109g and had to find our spring constant so that we could see the relationship between period and oscillation. First we weighed our mass and found it to be 12 grams which meant that only 4 grams went into the mass for oscillation. This also meant that we had to attach 105 g on our spring so that every one would be using the same amount of mass but different spring constant. Next we had to find our spring constant. To do so we set up our motion sensor to get the change in y with two different masses. We found that the position of y was .330 meters with 100 grams attached and .229 meters with 50 grams attached. To find the spring constant we used the formula …. change in force/ change in y… force was equal to Massxgravity and change in y was =.101  meters. These numbers gave us a spring constant of K=4.851  We put these numbers on the board along with every one else's. We also found the period of one oscillation using the 109g mass by using logger pro. 

What we did with this information: We plotted these numbers in an xy graph in logger pro and applied a power fit to get what we were looking for. In the x axis we put the period in seconds of each group and in the y we put the spring constant.

Conclusion #1: We Found that the relationship between period and the spring constant is that the bigger the spring constant the the faster the period and vice versa.

Part 2:

What we did: Next we got the period of oscillation using our spring for our original 109 g, 209g ,209g, and 409 g. To find the period we used the formula. Change in time(s)/ Periods. 

this gave us this chart.

IN GRAMS             period in seconds

109                                .9625

209                                1.35

309                                 1.56

409                                 1.77

Finally we plotted this and evaluated the graph

Conclusion: We found that if you use the same spring with the same spring constant that as you increase the mass to the bottom of the spring that the period of oscillation also increases.

December 4,2014. Moment of Inertia of a uniform triangle about its center of mass

Purpose: To be able to calculate the moment of inertia of a uniform triangle about its center of mass.

Apparatus: We used a Triangle, disk, hanging mass, a sensor, and a piece of string.

What we did:  First we took the measurements of the triangle.

 Next we got the apparatus. We tied a string onto the apparatus  with a hanging mass on the other end. Then we got the string and put it over a pulley. Next we opened up the hose clamp so that the bottom disk could rotated independently from the top disk. After doing this we opened up logger pro and opened up our sensors. We used the Rorary motion sensor and used the setting to count 200 marks per rotation.  Finally we collected our data. The slope of these graphs were the angular acceleration/ deceleration. In order to get our alpha we used ( alpha of hanging mass rising + alpha of the hanging mass dropping and we divided it by two. This gave us an average alpha to use in our calculations. 

Part two: We mounted the triangle in a vertical position and used the same technique to find alpha for the apparatus to find the alpha of the apparatus with the triangle.

Next we mounted the triangle horizontally and used the same technique to find alpha for the system while the triangle is horizontal.

Finally we used all the alphas to calculate the moment of inertial of each situation based on the hanging mass and the angular acceleration (alpha) In order to get the inertia of the triangles alone we had to find the inertia of the system with the triangle and subtract the inertia of the system with out the triangle.

Below are all of our calculations

INERTIA

Conclusion: Using all these different formulas we were able to get the actual inertia of the triangle in both positions. My lab partners Isaac and Mathew had a major role in this Lab. They guided  and showed me how to get all of these calculations.

December 3, 2014: Conservation of Linear and Angular momentum

Purpose:  Prove that linear and angular momentum is conserved.

Apparatus: We used the model ME-9281 Rotational Accessory Kit and the ME-9279A Rotational Dynamics Apparatus.

used this to find the velocity of the ball when it reached the end of the ramp

Used this to the angular speed of the ball once it is in the ball catcher.

What we did: We set up the apparatus in picture one. We picked an initial launch point for the ball to roll down from and marked it on the ramp. Next we made a few measurements. We measured the height of the launch point to the bottom of the apparatus(19.2 cm), the height from the bottom of the apparatus to the floor(97.5) cm, and the horizontal distance the ball traveled once we let it go from our launch point(51 cm). We used kinematics to solve for the velocity at the end of the launch ramp which is known as the launch speed.(1.1433 m/s) This Velocity will be used in further calculations.

Part 2:  We used the aluminum top disk and mounted the ball catcher on top of a  small torque pulley. We used logger pro so that we could find the acceleration and deceleration of the torque pulley.  

The slope is equal to the acceleration/deceleration

We used the mass and the average angular acceleration to calculate the inertia(I) of the apparatus.

Part 3: Finally we set up the ramp and measured the radius at which the ball would hit the ball catcher. We released the ball from our initial release point. This would allow us to use our Velocity at launch in our next calculation. Finally we did the calculations to find the angular speed.

How we found I

How we got our angular speed.

Conclusion: We found that our Omega was equal to 1.74 rads per second theoretically and that our actual value for omega was 1.69 rads per second which is real close. Reasons for error may be rounding errors, went slightly higher than our initial drop point, or made have a small measurement error.

November 26, 2014:Angular Acceleration

Purpose: Analyze how different masses, different disk of different diameters , and different torque pulleys affect angular acceleration.

Apparatus: This device is known as the Pasco Rotational sensor.

What we did: First we had to take the measurements of several items to at least 3 significant figures.

• Diameter and mass of top steel disk: 126.6 mm, 1356 g

Diameter and mass of bottom steel disk: 126.6 mm, 1348 g

Diameter and mass of top Aluminum disk: 126.6 mm, 466 g

Diameter and mass of the smaller torque Pulley: 12.5 mm, 10 g

Diameter and mass of the larger torque Pulley: 25 mm, 25 g

the mass of the hanging mass supplied with the apparatus: 24.5 g

Second we opened up logger pro. Next we set up the pasco rotational sensor to 200 counts per rotation. Then we opened up the hose so that the bottom disk would be able to rotate independently form the top disk when the pin was plugged in its place. Next we wrapped the string around the metal plate as the picture above shows to position the mass at the highest point on the pulley. finally we turned on the air and captured the and collected the data onto logger pro. This gave us an angular velocity vs. time graph which we were able to use to find the angular velocity of the upward and downward motion.

We used this same process to find how the change in the hanging mass effects the angular acceleration, the change in the torque effects the angular acceleration, and how changing the rotating mass effects the angular acceleration.

Conclusion: The lab asked us to analyze three thing which were outlined in the hand out

EXPTS 1,2, and 3 Effect of changing mass: We found that the heavier the mass the larger the angular acceleration. 

EXPTS 1 and 4 Effecs of changing the radius and which the hanging mass exerts a torque:  We found that the larger the radius of the pulley the larger the angular acceleration

EXPTS 4,5, and 6 Effects of changing the rotating mass:  We found that the lighter the roraring mass, the larger the angular acceleration.

November 19,2014: Moments of Inertia

Purpose: To use the concepts of inertia to predict how long it will take for a 500 g cart to roll down a 40 degree decline track for 1 meter.

Apparatus: Large metal disk on a central shaft, a cart, ramp, and a camera. We must find the Moment of inertia of this apparatus.

Pic of apperat

What we did: First we took the measurement of all the desired pieces. We got the diameter of the large metal disk which equaled 20.04 cm, the length of the two small rods were 5.15 cm each, and the thickness of the large disk was 1.56 cm wide.Next we set up a camera to obtain a video capture of the disk as it decelerated. We made sure to give the disk a push soft enough for it to spin 2 revolutions. We captured the video and opened it into logger pro. We first set an axis. Then we plotted the points as it revolved around in circles until it came to a complete stop. Finally we scaled the disk and made the diameter of the disk our scale of 20.04 cm. 

What we did with this info: This video capture and points gave us a graph and we were able to get the velocities in the x and y direction. This enabled us to get the final velocity which equaled (Vy^2 +Vx^2)^.5. Then we made a linear fit which gave us a slope that turned our to be our acceleration. In this case our deceleration. It was equal to - 2.385 . We were able to find alpha with this since we now had an acceleration and the radius. Alpha=AxR  This allowed us to come up with the formulas below to find the moment of inertia. With the calculations below we were also able to predict how long it would take for the cart to travel 1 meter.

Conclusion: Our Pridiction was that it would take 7.83 seconds. 

This is a picture of three different times it took for our cart to travel 1 meter. The Average was 7.84 seconds. This gave us an error of less tan .128%.

November 17th 2014: Predicting the Height of a Meter Stick After a Collision.

Purpose: The Purpose of this lab was to figure out the height that the meter stick would go in a rotation system after it collides with clay and sticks onto the stick for some set height.

Apparatus:We used iron rods and clamps to give us a foundation for our meter stick and our clay to stand on. Next we got a rotary sensor and attached it to the rod using clamps. This sensor was not actually used to obtain data. It was only used since it gave us a good post to slide in our meter stick which allowed it to rotate. We Then grabbed a computer and plugged in a camera to obtain a video capture.

Our Set Up

What we did. First we weighed the meter stick. Then we crimped down our meter stick with the gasket onto the iron rod.. Next we got a piece of clay, wrapped it around a piece of tape and weighed it. Next we made a prediction of how high it would elevate.

Our Calcuations Say H=.108

After making our prediction we had to see how high it actually went. We Placed the clay on the bottom rod, lifted our meter stick to make it horizontal to the floor, we pressed video capture, and finally we let go of our meter stick. The camera took video of the whole rotation

.Next we opened up our video on logger pro. First we set an axis on our video which was where the collision took place and horizontal to the floor. Next we scaled our video so it could get a measurement of how high it went. After we clicked plot and followed the movement as it hit the clay all the way to the peak height. This last dot gave us the final height.

Conclusion: We were able to predict the height the meter stick would travel using the formulas above. Our prediction was that it would travel .108 meters high. Logger pro stated that it traveled .08 meters high. Our prediction was off by 2.8 cm which is significantly close to the actual. Things that may have caused this uncertainty may be friction,.

November 6,2015: Conservation of Energy

Purpose: The Purpose of this lab is to determine that Momentum and energy are conserved during a collision.

Apparatus:

 Camera for our video capture

Table for our collision 

What we did: In order to obtain the information that we need, we first had to get a video capture of our two collisions. We First took a video capture of two steel balls. The first ball was at rest at the center of our table. We gave the second steel ball an initial push so that it could collied with the ball at rest. Next we did the same process of obtaining the video capture, but with one steel ball and the other being an aluminum ball. Lastly we measured the weight of all three balls which we would later use to analyze our data.

How we collected our data: We logged onto logger pro and pulled out our first collision from the file. This put us directly into video capture where we clicked add points. This allowed us to click the position of the ball that we pushed up to the collision and through the collision. These points gave us a position vs time and a velocity vs time graph. Next we picked add point again and we followed the ball at rest before the collision and after the collision. Then we picked the scale button and scaled the table that we did our collision on. Lastly we set the axis on the table. This last step gave us a smoother position vs time and velocity vs time graph in which we analyed.PPPPP

 What we did with our data: First we obtained our x,y coordinates in respects to time. Then we got the slope of the graph by applying a linear fit. This gave us Velocity initial x (Vix) and Velocity initial y (Viy). Next we got the slope of the graph after the collision by applying a linear fit which gave us the Velocity final x (Vfx) and Velocity final y (Vfy) We did this for both balls and obtained a Vf1, Vf2, Vi1, and Vi2 in both x and y. This allowed us to make new calculated columns. One was P total x= m1V1fx+m2V2fx. The second being P Total y=m1V1fy+m2V2fy. The third was KE total= .5m1(V1x^2+V1y^2 )+.5m2(V2x^2+V2y^2)




Conclusion. After our analysis of two collisions, we were able to determine that momentum and energy are conserved during a collision.Here Are Our Calculations which Prove this statement. These are relatively close to what the Blogger pro claimed the momentum in the x and y.

November 1,2014: Impulse Momentum Theorm

Purpose:The Purpose of this lab is to verify the impulse momentum theorm. Momentum is equal to P=MV. Impusle is equal to the integral of the force in respects to time. Together  the Change in momentum is equal to impulse.

Apparatus: We used a track, two carts, a motion sensor and a force sensor. We attatched a force sensor on the back of one cart. The other cart had a peice of plastic attached to a spring which would obsorb the collison. The force sensor measured the force between the collisons and the motion sensor measured the velocity of the cart.

What We Did: We opened up logger pro and imputed our two sensors so that they could be read. We zerored out our force sensor and changed the direction of our motion sensor. Now we were ready to do our collisions. We left the cart with the plastic stick sticking out stationary. Next we gave the cart with the force sensor a push so that it could collied with the stationary cart. after the collision the cart that was pushed toward the stationary cart was pushed back by the plastic stick with the spring.

Our Data: Once we collected our data by collecting it in logger pro, we had to evaluate it. In order to find the impulse, we had to integrate over the period of the collison(the force). This integration is shown as a Red solid in the graphs below. Our data shows three graphs. One is a force vs time graph, the second is a position vs time graph and the third is a velocity vs time graph.


In order to prove that the momentum-Impulse theorm is true we had to set up the equation below.

mVf - mVi= Impulse     Where the impulse is simply the integral of the collision,

m(Vf-Vi)=Impulse

Above is the collision between the carts with the mass of the cart being .99kg
m(Vf-Vi)=Impulse

m=.99

vi=.428

vf=-.371

-.79=-.753

Above is the collision between the carts with the mass of the cart being .435 kg

m(Vf-Vi)=Impulse

m=.435kg

vi=.268 m/s 

vf= -.245 m/s

-.22315=-.2405

Above is our experiment of when our cart collied with the clay: This causes Vf to be equal to 0 

m(Vf-Vi)=Impulse

m=.435kg

vi=.728

vf=0

-.31=.277

Conclusion: Our Data above does have a bit of error. However, the error is so low which allows this theory to be true. Reasons for error may have been false readings by our motion or force sensors. Overall we conclude that The Change of momentum is equal to T

he impulse where the impulse is simply the integral of the collision.(The force)

m(Vf-Vi)=Impulse

October 20, 2014: Magnetic Conservation.

Purpose: The purpose  of this lab is to prove that the integral of magnetic force with respect to time is equal to the change in momentum.

Apparatus:We used an air source to blow air into our triangular track which makes the track friction-less. We used a red cart to move up and down the track. We also used a motion sensor to measure the distance between the motion sensor and the metal clip board. A magnet was also placed below the motion sensor so that their would be a repelling force between the magnet below the sensor and the magnet attached to the end of the cart.

 Track,cart,motion sensor, and magnet

Air sourse

What We Did: In Order to get the information we needed to derive a formula to prove that the integral of the force was equal to the change in momentum, we had to obtain a relationship between the distance of the cart to the magnet and the angle of the track. First we raised our track to a small incline and turned the air source on so that the cart could begin to slide down the track. it eventually came to a stop where the magnetic forces repelled and we turned off the air source so that the cart could stay in place due to friction. In order to get the angle of the incline we had to download an app from our phone that obtained angles to a 10th of a degree of accuracy. This made calculating the angle so much easier since we only had to place the phone onto the track and it would give us the angle we needed. Then we measured the distance between the magnet below the motion sensor and the magnet in front of the cart. We Did This processes 9 times and came up with a chart similar to the one below. We Obtained a smaller distance as we increased the angle of the track.

CM     Angle (In Degrees)   Mass of the cart 

3.3      2.35                                     356 grams

2.6      4.6

2.2      5.7

1.9      8.2

1.8      8.7

1.6      10.6

1.4      12.4

1.3      15.8

What We Did With This Data: We logged onto logger Pro and began to plot our information in an x,y system. Our X= the distance between the magnets. Our Y=The force, Where Force=mgsin(theta) This gave us a graph in which we had to insert a POWER FIT to connect our data points.

Next we found a formula for the separation between the motion sensor and the clip board by using total distance-X. Next we made a formula for UR which was the integral of the force. This equaled (.0003785/.792)*Position^(-.792). We got this integral from the formula above which used the power fit and gave the equation Ax^B. We plugged in the numbers and integrated. We then made a new calculated column for kinetic energy which equaled .5MV^2. Finally we made a new calculated column for total energy which equaled Ur+KE

BELOW IS OUR GRAPH OF THE INFORMATION ABOVE

Conclusion: As our chart above shows, total energy is constant and ranges beteween .30 and .50 This Shows that  that the integral of magnetic force with respect to time is equal to the change in momentum.

October 5th, 2014-

Purpose: To be able to prove the law of conservation of energy.
Apparatus: We used a spring, a mass,  a motion sensor and logger pro.

Our set up

What we did: We grabbed a mass that was attached to a hook which was a total of .5kg. Next we grabbed a spring and weighed it which gave us .1kg. We set up by hanging the spring with out the mass stretching it and measured the height from the floor to the bottom of the mass.(76cm) Next we measured the stretch of the spring by getting the distance to the floor to the bottom of the mass.(55cm) This gave us a delta x of 21 cm. We Then we found the K in the PEe=.5kx^2 formula by using the formula Mg/Delta x which gave us 23.53 After we set up our motion sensor and activated it onto logger pro. We let the mass start to oscillate and then we began to record the movement of the mass/spring system for ten seconds. This gave us two graphs on logger pro which consisted of a position vs time chart and a velocity over time chart.

What we did with this information: With this information we added 5 new calculated columns. These included gravitational potential energy(GPE), Kinetic energy (KE), Elastic potential energy(PE elastic),gravitational potential energy of the spring(GPE RED),  Kinetic energy of the spring(KE spring) and a final column for the Total Energy(Total. we set up a formula of each column as followed.

GPE: Mass*gravity*position

KE:.5*mass*velocity^2

GPE RED: (Mass of spring/2) *gravity*postion

KE Spring: .5*(mass of spring/3)*Velocity^2

PE elastic: .5*K*Delta Position

Total: Sum of all forms of energy

We finally graphed all of our new calculated columns together in out position over time chart which gave us multiple graphs that looked as followed.

Conclusion: The image above proves that energy is conserved. The yellow line above is the total energy of the spring as it oscillates up and down with the mass. As you can see, the total energy is relatively constant between .6 and .7 J.

October 9th, 2014- Work-Energy Theorem

Purpose: The purpose of this lab was to find the relationship between work and energy. We had to prove that Kinetic energy is equal to total work done.

Apparatus: We set up our apparatus which consisted of a cart, a track for the cart, a spring, a force sensor, and a motion sensor.

Our set up

What we did: We logged onto logger pro and connected both of our devices which included the motion sensor and the force detector. We first zeroed out our force sensor. Then we connected our motion sensor and reversed the direction of the motion sensor since the cart would be traveling away from the motion sensor rather than toward. Next we pulled the cart and pressed collect data on logger pro and let go of the cart. This gave us two charts. One was a position vs time chart and the other was a position vs force chart.

What we did with our data: The first thing we did was add in an extra curve for Kinetic energy. That is the purple curve above. In order to prove that kinetic energy was equal to force we had to find the integral of the force which is the red line above. We did this by clicking the integral function in logger pro. We integrated at three different locations. Below are our three charts already integrated.

Conclusion: The Graphs above prove that Kinetic energy is equal to the integral of force. This proves that kinetic energy is equal to total work since the integral of force is force over a specific distance. You can check this by looking at our graph and comparing the kinetic energy and the integral for force. This is extremely useful when finding energy problems and you are also given a force.

October 5th, 2014-WORK AND POWER

Purpose: The purpose of this lab was for us to learn how to calculate work and power. We did this by performing an outdoors activity and calculating our results.

What we did: Our lab asked us to find the amount of work and power it took for us to climb up the stairs. The first thing we had to do was to find the height of the stairs. to do so we measured the first step which was 17cm and multiplied it by the amount of stairs.(26 steps) This gave us a height of 442 cm or 4.42 Meters. Next we found the amount of time it took for us to walk up the stairs and to run up the stairs.  Below are my results

Time it took to walk up the stairs: 15.63 seconds
Time it took to run up the stairs: 5.03 seconds 

Part Two: The second part of the lab asked us to find the work and power it took for us to pull a bag from the floor up a pulley system. First we determined our height which was equal to the height of the stairs.(4.42Meters) Next we had to pick one of the three bags. I chose the bag that weighed 5 kg. Next one of my lab partners went up the stairs to calculate the time it took for me to pull the bag all the way up the pulley system. It took me 4.42 seconds.

What we did with this information: We used this information to calculate the amount of work and power it took for us to pull the bag up, walk up the stairs, and run up the stairs.

Work for the stairs project: We used the formula Work=force*distance. In order to calculate force we had to find our mass. I Converted my weight in pounds into kg. I weigh 170 pounds which converts to 77.11 kg.
1)work for walking and running up the stairs
W=Mass*acceleration*Height
W=77.11*9.8M/S^2*4.42M
W=3340.1 Joules.

2)Power for walking up the stairs:
Power=Work/Time
power=3340.1J/15.63S
Power=214.14 Watts

3)Power for running up the stairs:
Power=Work/Time
power=3340.1J/5.03S
Power=665.41 Watts

4)Work to pull mass up the pulley system:
Work=force*distance
Work=5kg*9.8M/S^2*4.42Meters
Work=216.5 Joules

5)Power needed to pull up mass
Power=work/Time
power=216.5J/3.5seconds
Power=61.88 Watts

Conclusion: We were able to calculate work and power in an everyday activity. This lab showed us how to approach any work or power problem. The main formulas used were Work=Force*Distance and Power = Work/Time

September 1st, 2014- Relationship between angular speed and Theta

Purpose: The purpose of this lab was for us to find the relationship between omega and theta using a conical pendulum.
Apparatus: 

This is our conical pendulum. This apparatus has a pole attached to the top of it and a string tied to the end of the pole. It is hooked up to a motor which cause the pole to pin in a clockwise direction. 

What we did:  First we took the measurements of the  few items we could obtain using a meter stick. We found that the Height of the Conical pendulum was 211 cm and the length of the string was 165 cm. Next we took the time for the weight at the end of the string to revolve around the apparatus 10 times. Once we got the time we recorded our data. After we had to determine the height of the mass by setting up a tripod and attaching a piece of paper onto it. To get the height we would have to position the tripod close enough for the weight to hit it and elevate the tripod to the point where the mass just hits the top of the paper.

Once the mass hit the paper we got the height of the paper relative to the ground by using a meter stick and recorded our data. That completed the experiment for one velocity. Professor Wolf would then raised velocity and we would obtain the same data for that speed. We continued this process for 7 different speeds. This is what we obtained

TIME FOR 

TEN ROTATIONS------HEIGHT OF MASS AT THAT VELOCITY

37 SEC                           49CM               

30 SEC                           72CM

26SEC                            92.3CM

25 SEC                          104CM

23 SEC                          119 CM

21 SEC                           132 CM

19 SEC                           174CM

WHAT WE DID WITH THIS INFORMATION: First we had to come up with a formula to be able to use our data. We set up an image of the apparatus and had to find out a few unknowns.

H= height of the apparatus.=211cm

L=length of string.

H2= the the height of the mass while spinning= column 2 above

H-H2=to the height of the string to the top of the apparatus.

Theta= acos(H-H2)/L 

We then made a body diagram of the mass and the tension. We separated the tension vector into x and y components. We set the x component= m*(V^2/R). We then found R using the Pythagorean theorem. Finally we manipulated and solved the equation to find the relationship between omega and theta.                                BELOW ARE ALL THE STEPS MENTIONED ABOVE.

Next we opened up Logger Pro and Inserted all the information we obtained.(Below)

We Then Graphed it and put a linear Fit (Below)

Conclusion: After an extensive lab we were able to find the Relationship between Omega(W) and Theta by finding the Height of the pendulum,the length of the string, the time it takes for the mass to revolve around the pendulum, and the height for that time period. We found that the relationship is that tan(theta)=(Omega^2*R)/G.