April 17, 2017 Lab 13: Magnetic Potential Energy

Lab 13: Magnetic Potential Energy

Chris Ceron, Amy Chung, John Choi

Goal of Lab

Derive an equation of magnetic potential energy using different cases and quantities of kinetic and potential energy.

Theory/Introduction
Unlike Gravitational or Elastic Potential Energy, we do not have an equation for Magnetic Potential Energy. However, we can derive an equation based on what we know about conservation of energies. At the moment that Kinetic Energy is zero, the energy within the system is all transferred to Magnetic Potential Energy to repel our object. Our apparatus allows for this to occur.

Air track and glider

We angled the air track to seven different measurements. As the angle of tilt for the air track gets larger, the glider gets closer to the repelling magnet. With the data acquired we plotted a graph on LoggerPro and applied a power fit to find the equation of Magnetic Potential Energy.

For any system with a non-constant Potential Energy, the Potential Energy "U" is caused by an interaction between force F. We can mathematically establish that the relationship is:

r = separation distance

We can also conclude that the potential energy in our system as the distance between the repelling magnets get infinitely further away is 0.

Using a free body diagram, we found that at equilibrium, the force of the magnet is equal to the x component of gravitational force.

Once we have our equation for Magnetic Potential Energy, we can verify and test the conservation of energy within our system. We placed our glider on the opposite end of the magnetic and provided an initial velocity, which also gives Kinetic Energy. At a certain distance, the Kinetic Energy is all transferred to Magnetic Potential Energy. As the magnets repel and move the glider backwards, the energy is transferred back into Kinetic Energy.

Procedure

We started the procedure by leveling the track. Then we increased the angle and tilt of the track by adding a book on the opposite side of the magnets. For each increment, we gathered the different angles and the separation distance between the two magnets. After seven increments, we plotted Force vs. distance and applied a power fit to derive an equation F(r). As mentioned in the Theory/Introduction, we could integrate the F(r) to find U(r).

After deriving our U(r) function, we can verify the conservation of energy within our system. We first placed our glider closer to the fixed magnet when the air is turned off. To find the separation between the two magnets, we recorded the distance between the motion sensor and the glider.

Once we have our separation value, we place the glider on the opposite end of the track. With the air turned on, we gently push the glider and use our motion sensor to measure the distance and velocity. Through LoggerPro, we create a column for Kinetic Energy, Magnetic Potential Energy, and Total Energy of the system. We use a single graph to display each column, which allows for us to verify the conservation of energy within our system.

Data:

Angle (degree)	separation (m)
2.5	0.0239
3.8	0.0232
5.1	0.0212
6.4	0.0199
7.9	0.0183
9.1	0.0176
10.5	0.0163

A phone application was used to measure the angle of tilt of the air track. We used both the long and short side of the phone to measure the angle, and took the average of both.

Uncertainty of angle measurement: +/-.01

Mass of glider: 0.3471 kg

Initial Separation was 0.017 m

As mentioned in the Theory/Introduction, we used our data to plot a graph of Force vs separation (r). We could then apply a power fit and derive a function F(r).

For function F(r) came out to be:

Our uncertainty for A was 7.149e-6 (also equals 47.9%)

Integrating our function gave us

Making a calculated column for Kinetic Energy, Magnetic Potential Energy, and Total Energy and putting them in one graph gave us:

We see that the total energy within the system is not necessarily constant. 

Conclusion
Although our results had a large uncertainty, we were able to utilize our apparatus and data to show that energy is conserved within our system. In the graph above, we notice that the total energy of the system before and after the magnets repel each other is in the form of Kinetic Energy. Some sources of error that caused a large uncertainty were the placement of the magnets. For our apparatus, the magnet had to be adjusted and could have been aligned incorrectly. Throughout our lab, we could see the magnets slightly shift.

For the second portion of the lab, the separation value we used could have greatly affected the outcome of our graph. Even a minute change in the separation value would cause a large uncertainty with our Magnetic Potential Energy.

April 10, 2017 Lab 11: Work-Kinetic Energy Theorem Activity

Lab 11: Work-Kinetic Energy Theorem Activity

Chris Ceron

Goal of Lab

Prove the work-kinetic energy theorem by comparing the calculated value of KE and comaparing it to force times distance

Theory/Introduction

We define work to be a force multiplied by a distance:

The work-kinetic energy theorem states:

In this lab, we tested different experiments to compare the calculated value of work and comparing it to the value returned through LoggerPro.

Experimental Procedure

Experiment 1:  Work Done by a Constant Force

For experiment one, we used a level track for a frictionless cart that was attached to a hanging mass. The hanging mass also went through a frictionless pulley system.

Our apparatus also includes a force and motion sensor, and a stopper for the cart.

Once the sensors were calibrated, we attached the force sensor to the cart, and attached the hanging mass to the force sensor using a lightweight string. We placed the motion sensor behind the cart to give us the displacement vs time of the cart while it is being pulled by the hanging mass. Through a free body diagram we found that the tension on the string was equal to the hanging mass times gravitational acceleration. We used this tension force and multiplied it by the displacement to calculate the work done.

We also found the change in kinetic energy by using LoggerPro to find the velocity of the cart. From there we used our kinetic energy formula:

Experiment 2: Work Done by a Nonconstant Spring Force
For experiment 2, the work done changes based on the displacement of the cart. The cart is attached our force sensor using a spring.

The force of a spring is:

To find the value of k, we recorded the cart when the spring was unstretched to stretched to a known distance. LoggerPro produces a Force vs. Distance graph, and this is used to calculate both the value of k by using a proportional fit and the work done on the cart. To calculate work, we used the change of the spring's potential energy:



The spring's initial potential energy is 0, so the final equation used was:

Experiment 3:
For experiment three. we stretched the cart to the same known distance as experiment two. We recorded the changing values of force vs. distance as the spring pulls the cart and the changing values of kinetic energy. These values are predicted to be the same. Similarly to experiment two, we use LoggerPro to find the area under the graph of force vs. distance and compare the two values of work. 

Experiment 4:
For experiment four, we watched a video in which a professor uses a machine attached to a rubber band. The set up in the video included a force reading, photogates to measure the displacement, and time intervals. We used the results in the video to calculate work by multiplying the force times the distance.

Data:
Experiment 1:
The hanging mass in experiment one was .050kg
The total mass of the cart was 1.19kg.

Our initial readings for experiment one. Shows a constant force.

We manipulated the force graph above to also display kinetic energy by creating a new column that multiplied the mass (constant) and velocity of the cart based on the sensor's recording. 

Force graph also displays kinetic energy

We changed the time axis from three to one and found the area under the force graph using LoggerPro's integral feature. We then displayed the value of kinetic energy for our highlighted section and compared the value to the area under our force graph.

Displays the values for kinetic energy and the area under the graph for the highlighted section

The value we received for the area under the graph was 0.1659 N*m

The value we received for the change in kinetic energy for the same section was 0.156 J (J = N*m)

The two values are relatively close.

Experiment 2:
As mentioned in the Theory/Introduction, we first found the value of k by recording the cart when it was pulled to a known distance. The distance we stretched our cart to was 0.6 m. Through LoggerPro, we could find the slope of the graph to give us our value of k.

The value of k we obtained was 3.460 N/m.

Using this graph, we could also find the area under the graph to find work.

Work done by stretching the spring

The value we obtained for work was 0.5146 J

We also calculated our work using the elastic potential energy formula

Experiment 3:

The following graph shows a specific point where we compared the area under the Force graph vs. position to kinetic energy. The value for the integral of the force graph is negative due to the positioning of our motion sensor, so we compare the magnitudes.

Area under the graph = -.6583 N*m. Kinetic Energy of the same section = 0.675 J

 Experiment 4:

For experiment four, are calculations were based on splitting up sections of the graph into simpler calculations of triangle, trapezoids, and rectangles. 

For triangles, we used:

For trapezoids, we used:


For rectangles, we used:


Calculation:


Using the displacement and time provided in the video, we can calculate our final velocity, and final kinetic energy:




The two values are relatively close.

Conclusion:
The work done on the cart by the spring are relatively similar to its change in kinetic energy. As the spring exerts a force on the cart, the cart continues to increase in velocity. Similar to the experiments, you can find any two points while the spring exerts a force on the cart, and use the velocities at those points to find work. Work is the change in kinetic energy according to the work-kinetic energy theorem, and the change in velocities come from the spring's force accelerating the cart.

Overall, the data followed our expectations as the area under the graph of force vs. distance was very similar to the kinetic energy. Some uncertainties that caused our numbers to slightly differ was the calibration of our force sensor. We calibrated the sensor more than once because it was recording incorrect values when tested. Another uncertainty could have also been the motion sensor. The sensor will not accurately record the carts distance if the cart begins its displacement too close to the sensor. This will cause a fluctuation in our velocity, which results in it affecting our kinetic energy.

Although these uncertainties existed, they were only slightly present. Our numbers in each experiment were very close and we could easily conclude that the area under the graph of force vs distance and kinetic energy are the same value.

April 03, 2017 Lab 10: Work and Power

Lab 10: Work and Power

Chris Ceron, Amy Chung, John Choi

Goal of Lab

Perform various physical activities and record force, distance, and time for power

Theory/Introduction

Work in relation to force is:

F = Force

 = change in displacement 

Power is the relationship between how much work is done in relation to time:

Experimental Procedure

We did three activities to find our power output.

1) Lift a known mass up by a measured distance

We used a pulley system to pull a backpack with a known mass up to the second floor balcony

Through a free body diagram, we were able to find an equation for the Force we used to pull the backpack up:

To find work, we multiply this force by  (the displacement of the backpack from the ground to the balcony)

We used this to find power

2) Walk up stairs and record the time taken to complete

3) Run up stairs and record the time taken to complete

These two activities are almost identical, except one involves walking and the other running. Through a free body diagram, we found that:

Similar to activity one, we found the work output by multiplying force times displacement

We used this to find power

Data:

time 1 (sec)	time 2 (sec)	time 3 (sec)
4.65	15.8	8.5

The height of one stair was 0.169 meters. There were 26 stairs, so the total height = 4.394 m.

mass of the backpack in activity one = 5kg

my mass for activity two and three = 86.2 kg

Power 1 (Watts)	Power 2 (Watts)	Power 3 (Watts)
46.3	234.9	436.7

Conclusion:

a) There would be some measure of kinetic energy. Assuming my final velocity was 1 m/s, my final kinetic energy would be:

My work from gravitational potential energy was:

The ratio from KE to GPE is 43.1/3711.9 which is .0116 or 1.2%. The amount of error is relavtivately small, but show how some quantities are ignored.

b) Assuming one flight of stairs is equal to the 26 stairs, I generated 436.7W running up one flight of stairs. 1100 Watts / 436.7 Watts gives us a ratio of 2.5. I would need to climb 2.5 flights of stairs to equal the power output of the microwave oven.

c)The power output of the microwave is 1100W per second. To get the total power output, you multiply this by 6 minutes of 360 seconds. That equals to 396000 Joules. The work done climbing one flights of stairs was 3711.9, so 396000/3711.9 equal to 106.7 flights of stairs.

d) 1) Power = Work/Time. So 12.5MJ/10 minutes or 600 seconds = 20833.3W

2) 20833.3W divided by 100W per person = 208.3 persons.

3) If you were the only person providing energy and you put in 100W, you would have to ride 208.3 seconds to hear water for your 10-minute shower.

The lab ignored several factors that would alter the results obtained. For example. we did not take into account the kinetic energy obtained from running up the stairs. In activity one, the force was exerted on the bag was not constant, and friction between our hands and the rope was ignored. This is a large uncertainty as there was little grip and a lot of friction between our hands and the rope. For this lab however, these factors were ignored to simply calculate a power output.