Lab 12: Magnetic Potential Energy/Impulse-Momentum Activity Lab

Physics Lab #12 started on 4/15/15 with Edgar and Kaz
Magnetic Potential Energy Lab & Impulse-Momentum Activity
Annemarie Branks
Professor Wolf

Experiment 1:
Objective: Determine the relationship between the the separation distance of two magnets and their potential energy.

Procedure:
     1. Set up your vacuum and air track so when you place a cart on it, it ends up at equilibrium. Record the angle at which the track is equilibrium. "The magnetic repulsion force should equal the gravitational force component on the cart parallel to the track."

     2.  Set the air track at various angles with the vacuum blowing out air. Measure the distance between the two magnets for each trial. Our mass of the cart is 0.354 kg +/- 0.001 kg. We want to know the Force of the magnets so we can find their potential energy equation. In the Free Body Diagram below, we can see that the Force of the magnets is equal to mgsinθ. 

This is our measured data. The angles have an uncertainty of +/- 0.1 degrees and the separated distance have an uncertainty of +/- 0.001 meters.

     3.  Using LoggerPro, make an x and y column. The x will be the distance between the magnets, and the y will be the Force of the magnets which was mgsinθ. Generate a graph with these new columns along with a PowerFit.

      4.  Since our data points are in a curve, we can assume the function is in this format: F=A*r^B. A and B have a error margin of +/- 10%. Our Function is  F=0.0002380r^(-1.848). Since Potential Energy is the negative integral of Force, we can come up with an equation for Potential energy.

     5.  Set up a motion detector on the stationary magnet side. Give the cart a push and begin collecting data on a Position vs. Time graph. Columns for time, position, and velocity should be made by LoggerPro. Create columns for the separated distance (which in our case was "position" - 0.20m), Kinetic energy, Potential Energy of the magnets (which we figured out in Step 4; r = "separation distance"), and Total Energy (KE + U). Generate a graph to include all of these columns in relation to time. 

Conclusion: We really want to just focus on the part of the graph where we see good data that proves Conservation of Energy. The change in direction of the cart is around two seconds. We can see how, just like in spring problems we have done, that the magnets posses a potential energy that goes to zero when the mass reaches its max velocity.  

     We now know that the smaller the separation distance between the two magnets, the larger the potential energy is. The graph tells us that when there is a very small separation distance the force is very big. You can visually see from pushing the cart toward the stationary magnet and it slowing down and flying back that there is a noticeable potential energy. 

Experiment 2:

Objective: Prove the Impulse-Momentum Theorem using a cart and spring. 

Procedure:

     1.  Calibrate a force sensor, and set up your system with the force sensor on top of your cart so that it will horizontally push into the spring you have set up with a clamp and rod. Use an L-shaped bracket and screws to secure your force sensor to the cart. Weigh the new mass of the cart. Our mass was 0.677 kg. You will also have a spring for the cart to run into. A motion detector should be placed on the opposite end of the track from the spring. For our experiment we set the motion detector to detect at 100 data points per second.

     2.  Using LoggerPro, collect data of your moving cart for a Velocity vs Time graph and a Force vs Time graph. The dramatic change in the graphs are where the collision occurred. Highlight and integrate the area under the curve on the Force vs. Time graph to find Impulse. Impulse should equal to the change in momentum, so find the final and initial velocity of the collision which are indicated by the vertical rectangle below. Multiply your change in velocity by the mass to get the change in momentum. Your change in momentum theoretically should equal to your impulse.

LoggerPro gave use an impulse of 0.4231 s*N and for the change in momentum we got -0.315 s*N. The final velocity was -0.237 m/s and the initial was 0.228 m/s. The difference of those velocities times 0.677 kg gave us our momentum. I believe the Force graph is upside down and give us a negative value. 

     3.  Repeat the experiment with added masses. We added 400 grams for a total mass of 1.077 kg. 

   Generate another graph like the one made in Step 2. 

LoggerPro gave use an impulse of -0.4523 s*N and for the change in momentum we got -0.402 s*N. The final velocity was -0.176 m/s and the initial was 0.197 m/s. The difference of those velocities times 1.077 kg gave us our momentum.

Conclusion: The first trial gave us an impulse that was supposed to be negative, but if it were then the value would be close to the change in momentum. We got even better results for the second trial where the impulse and the change in momentum were even closer. It possible to attribute these values to errors in system, measurements, and human discrepancies. The spring wasn't at a perfect angle, LoggerPro can only capture so many frames per second, and we may have been slightly off when picking the initial and final velocities of the cart. With all this in mind, the results were fairly good in demonstrating that Impulse and momentum are equal in a system.  

Experiment 3:

Objective: Prove the Impulse-Momentum Theorem using an inelastic system. 

Procedure:

     1.  Set up your system similar to how you did in Experiment 2, but replace the spring with a piece of clay attached to a wooden block. Make sure someone is holding down the block during the experiment so it doesn't move. At the end of the Force sensor there is a hook. Put the hook in one end of the a stopper and a nail in the other end of the stopper. Place the clay/block so the nail will stab directly into it. Make sure to get rid of any holes made in the clay before starting or repeating the experiment.

 

     2.  Give your cart a push and collect data to make a Velocity vs Time graph and a Force vs Time graph. Here are our results: 

     3.  Just as you did in Experiment 2, find the Impulse and change in momentum by finding the change in velocity and the Integral of the Force vs. Time graph. There was "shakiness" when the nail penetrated the clay. so we highlighted all of the graph where that occurred so we didn't exclude any energy transfer. LoggerPro gave us an Impulse of -0.3564 s*N. Our final velocity is 0 m/s and our initial velocity is 0.277 m/s. The difference between the two multiplied by the mass 1.077 kg gives us a momentum of -.0298 s*N. 

Conclusion: We wanted to reconfirm that Impulse and Momentum are equal even in an inelastic system. We focused on the part of the graphs where the collision occurred and found that the Impulse and Momentum calculated are fairly close in value. There are still errors, as mentioned in Experiment 2, that could attribute to the difference in the numbers. Still, this is a fairly good demonstration of the Impulse-Momentum Theorem. 

Lab 11: Conservation of Energy - Mass-Spring System

Physics Lab #11 started on 4/08/15
Conservation of Energy - Mass-Spring System
Annemarie Branks
Professor Wolf

Objective: Unlike most of the physics problem we have been doing, we want to prove the Law for Conservation of Energy with a non-negligible mass for the spring.

Procedure:
     1.  Set up a system so you have a spring attached to a hanging mass over a motion sensor. Downward will end up being the positive direction and the top will be the origin. In our experiment H=1.457m, y=0.857m, and L=0.6m for an unstretched spring.

     2. Find a spring constant by either using a force sensor like we have done in previous labs, or by weighing masses on the spring and measuring its stretch. We chose the second option which in hindsight probably wasn't the best option because we didn't get consistent k values for different masses. They were relatively close of each other so we just averaged them and got 8.92 k/m.

     3.  Put 250 grams on the end of the spring and create a Position vs Time graph and a Velocity vs. Time graph.

     4.  Since we want to include the mass of the spring, the gravitational potential energy and the kinetic energy are different from the typical equations we use; however, elastic potential energy stays the same since that equation does not include mass. This is how we came up with our new equations for gravitational potential energy and kinetic energy:

     5.  With our known masses and the known unstretched position of the spring, we can plug in these equation into a new calculated column and generate graphs. So for the "y" in the GPE equation we are going to use the "position" column to replace it. Use the "velocity" column to replace v-end in the KE equation. Finally, for the elastic PE's delta y, use (unstretched - "position").

   For our experiment the mass of the spring was 0.085 kg and the unstretched position was 0.857 m..

     6.  After plugging in our equations, we generated a graph. We also included a Total Energy line which was achieved by simply adding KE, EPE, and GPE together. The Conservation of Energy Law tells us that the Total Energy line should be straight.

You can see that wherever one energy rises another will fall and this proves the Work-Energy theorem. The reason the Total Energy line is not straight it because of some variations and errors as listed below; however, this is fairly good data to prove the Total Energy Theorem in a classroom.

Errors:

• An assumption we made in the lab was that the spring was uniform. So elastic energy of the spring would not be the equation we used for it 0.5k(x)^2.
• There was uncertainty in where "zero" should begin on the spring.
• There are some uncertainties in the measurements we took which is typically +/- 0.001m, but it was difficult to measure a hanging object that moved easily so the uncertainty is probably slightly more that 0.001 meters. 
• Vibrations and heat are also forms of energy, which could have some from the spring making contact with the bar from which it was hanging from, or even the spring wire hitting against itself and making a noise.
• I also mentioned earlier that there was uncertainty in our k value because we were not getting the same answer every time, just something that was close. If we had found the k value using the force sensor then there would be uncertainty from the force sensor. Better, more expensive equipment would reduce the uncertainty. This is typically the case for all devices like the motion detector and LoggerPro itself.

Lab 10: Work-Kinetic Energy Theorem Activity

Physics Lab #10 started on 4/06/15
Work-Kinetic Energy Theorem Activity
Annemarie Branks
Professor Wolf

Objective: Using Hooke's Law, find the spring constant for a spring attached to a cart on a track.

Part 1: Work Done by a Non-constant Spring Force
Procedure:
     1.  Set up your cart on a track  with a motion detector, force probe, and spring as shown in the picture below. Using LoggerPro, calibrate the force probe. You may want something underneath the spring to support it.

     2.  Using the file L11E2-2 (Stretching Spring) in LoggerPro, you will create a Force (N) vs. Position (m) graph. Make sure the motion detector is set to the Reverse Direction so when the cart goes toward the motion detector, it is travelling in the positive direction. Zero the motion detector and the force probe.

     3.  Stretch out the spring by pulling on the cart and collect data for when you release it. This is the graph we got from our cart and spring:

You can see by the extreme variation in the line, where the chart had its collision at the end of the track.

     4.  To find the spring constant for our experiment, highlight the part of the graph that is usable data and find the slope. We are doing this because of Hooke's law which tells us that the stretch of the spring is proportional to the force or, F=-kx. In our experiment we found that k= 1.759 +/- 0.01568.

     5.  Using the integration routine in LoggerPro, you should be able to find the area under the curve. This is how you find the work done in stretching the spring. 

Objective: Prove the Work-Energy Theorem by graphing and comparing Force vs. Position and KE vs. Position graphs.

Part 2: Kinetic Energy and the Work-Kinetic Principle

Procedure:

     1. Using the same set up as in part 1, we are going to compare Force vs. Position to Kinetic Energy vs. Position. In order to graph for Kinetic Energy we need to know the cart's velocity and its mass. So we measured the cart's mass, which was 0.505 kg, and we can find the changing velocity in various ways like (F/m)*t. We make a new column, for KE, in our data table with the equation 1/2*m*v^2, and graph it.

     2.  Click on the y-axis title to include both Kinetic Energy and Force versus Position on the same graph.

     3.  Using the Analysis Feature in LoggerPro you can find and compare the graph's position, force, kinetic energy, and integral (or work). Do this for multiple points.

For this highlighted area on the Force vs. Position graph, the area is 0.02664 N*m and the KE is 0.027 Joules (or N*m).

For this highlighted area on the Force vs. Position graph, the area is 0.07105 N*m and the KE is 0.069 Joules (or N*m).

For this highlighted area on the Force vs. Position graph, the area is 0.1100 N*m and the KE is 0.108 Joules (or N*m).

Though we could continue this for more areas, we can already see that the values for the areas under the curve are almost the same to the Kinetic Energy at that position. These values are not exactly the same due to errors in our apparatus. The graph should be a straight line with a y-intercept at zero. Considering the imperfection in this experiment, these graphs still prove that 1/2*m*v^2 = Work.

Objective: Using an alternative method as shown in the video, prove the Work-Energy Theorem. Discuss discrepancies in this method.  

Part 3: Work-KE Theorem

Procedure:

     1.  In LoggerPro there is a file called Work KE theorem cart and machine for Phys 1.mp4 with a video demonstrating how Force vs. Stretch graphs of a rubber band used to be made before programs like LoggerPro existed. The rubber band is attached to a cart which eventually passes through two photogates. This give the final speed of the cart which was can use to find the final Kinetic energy. As the rubber band is stretched, a marker is drawing on a large graph that the professor in the video is trying to move as a constant speed. You will see many variations within the lines due the human error, so the professor repeats the process until they get a good estimate of what the graph really should look like.

     2.  There are two methods to finding the area under the curve. The first method is weighing the paper, measuring the area of the paper, cutting out the area under the curve, weighing the area under the curve, and determining the area under the curve using a conversion factor. The second method is counting how many squares are under the curve and multiplying that number by the area of one square.

   Just to get an idea of the second method, we approximated the area under the curve by dividing it up into 4 different sections and making either triangles or trapezoids out of those areas, and finding the areas that way. By doing this, this is how our graph turned out:

   We found the area for each section, added them up and found that the total area under the curve was 26.595 N*m. This is how much work is being done in the system. Let's compare this value to the change in Kinetic energy, because according to the Work-Energy Theorem they should be equal. The mass of the cart was 4.3kg, the change in distance was 0.15m, and the time it took to travel from one photogate to the other was 0.045 seconds. With 1/2*(4.3)*(0.15/0.045)^2, the kinetic energy is 23.9 Joules.

   These answers are of course not completely accurate but they are relatively close to each other. Using calculus is the best method but it would be difficult. If we divided the graph up into smaller sections and used better, more expensive equipment we would find an answer with fewer errors. There are errors in the professor pulling on the graph paper and errors due to friction in the system. This is not a very accurate representation of the desired experiment but it was the best the professor could do with her limited resources at the time of the film's production. This experiment is not the best way to prove that the Work-Energy Theorem is correct.

Lab 9: Centripetal Force with a Motor

Physics Lab #9 started on 3/25/15
Centripetal Force with a Motor
Annemarie Branks
Professor Wolf

Objective: Determine the relationship between the the angle created by a revolving mass tied to a string, and the mass's angular speed.

The Apparatus:

A motor will spin the central shaft which in turn spins the meter stick. Attached to the meter stick is the string with the mass tied to the end. The string will make an angle, as shown in the picture.

Procedure:
     1.  First, measure the length, L, of the string, the internal radius, R, the apparatus makes with the end of the meter stick, and how high, H, the meter stick is above the floor from where the string is attached. For our experiment, the apparatus was 2.00 +/- 0.001 meters tall, the radius was  0.870 +/- 0.001 meters, and the string was 1.664 +/- 0.001 meters long. 
     2.  The first trial will require a certain number of voltage to give the system some angular speed. After making a mark on one point of the circle, time the system for 10 rotations.
     3.  Near the edge of the circle use a ring stand and a horizontal piece of paper that you will slowing raise up so you can find where the mass hits the paper. Once the mass hits the paper, find the height, h, the paper was at when the mass hit. 

     4.  Repeat steps 2 and 3 using more voltages for each trial than the last, until you have a total of 6 trials.
     5. In each trial you found a height of the mass from the floor, h, and an angular speed which you find from multiplying the number of revolutions you timed for by 2*pi, and divide that value by how much time it took to complete those 10 revolutions.
     6. Draw a free body diagram with all the forces on the mass. Calculate the sum of forces and find the relationship between angular speed and the angle of the string. Here are our calculations: 

     7.  We don't know theta, but we know other values that will help us find theta as shown in the picture above. The only unknown variable leftover is h. We want to be able to compare the values we got for angular speed in step 5 to the angular speed we will get from the equation we just calculated. If they are similar and we plot them on a graph then the graph should be directly proportional and the slope should be about 1 radians/second^2.
     For my group's experiment, we graphed the angular speed that we got from the equation above by the angular speed that we calculated in step 5. Our slope, A, was 1.006 +/- 0.006427. Just for fun, we flipped the axes and got a slope of 0.9935 +/- 0.006345.

Errors:

Obviously, there will always be a certain amount of uncertainty in ever measurement we take and even tough we produced nearly perfect results we must account for the uncertainty. 

• For the Length of string, the internal radius, and the height of the apparatus, we said they each had an uncertainty of +/- 0.001 meters. 
• For the height of the spinning mass above the ground we said there was an uncertainty of +/- 0.005 meters for each trial except for the last one, because it was a spinning so fast it was had to see where it hit. For trial 6 we said the uncertainty was +/- 0.01 meters. 
• There was also uncertainty in the reaction time of whoever was timing the spins. In our group we had two people timing the 10 revolutions and some times the people timed it more than once. For every trial the times were always within about 200 milliseconds, or less, of each other. Some online sites say that this average human reaction is slightly above 200 milliseconds, but since were young people with better reaction time than people of other ages, 200 ms is a good uncertainty for time.

Changes in the graph:

The uncertainties mean that the "real" value of the points could be up/down and left/right a certain amount. I wanted to focus on one point to show what all the possible values could be. I am keeping all calculations with the proper amount of significant figures. 

• The very first trial gave us 10 revolutions in 37.67 seconds which is 1.668 radians/second. If  we include the uncertainty of 200 milliseconds, the values are 1.677 radians/second and 1.659 radians/second. If these values for angular speed were on the y-axis, the values are within 0.009 radians/second up or down.
• Using the height that we got from the first trial as well as the height of the apparatus and the length of the string with all of their uncertainties, we are going to find a range for the x-axis. Without the uncertainties, the angular speed  is 1.67 radians/second. 
• Rather than doing the propagated uncertainty method that we have been doing in previous blogs, I used the small H, big h, big L, and small R values to find the largest the angular speed could be. I did the opposite to find the smallest angular speed. These ended up being 1.68 radians/second and 1.65 radians/second. If these values for angular speed were on the x-axis, the values are within 0.02 radians/second up or down. Reminder, I kept this in the proper amount of significant figures so 0.02 is not an exact number.

One could repeat this process for all the points but the general understanding is that the value we calculated has a range around it in the x and y directions for what the real value is. The real values should line up to give a slope of 1.