Magnetic Potential Energy Lab

Objective: 

• To come up with an equation to calculate magnetic potential energy
• To verify that the theory of conservation of energy applies to this system.

Part 1:

• Finding an equation for magnetic potential energy

Set-up:

1. Let an air track sit on the horizontal table.
2. Place the glider on the air track, make sure the air track is leveled which means that the glider stays still unless we give it a push.
3. Attach a magnet at the end of glider, and another magnet with the same polarity at the end of air track as well.
4. Setting up a motion sensor behind the magnet on a vertical rod, make sure the sensor can read as the glider moves. To make it easier for the sensor to read, we ask a square on top of the glider.
5. Connect the other end of air track to the air source which is already connected to the power.

The apparatus of the experiment is shown in figure 1

Figure 1: The apparatus of the experiment.

Performing the experiment:

• We use books to raise one end of the air track and turn on the air source to let the glider move on the air track, then give the glider a push (Figure 2)
  
  Figure 2: Using books to tilt one end of the air track.
  
  
• A glider will move toward the magnet and when the glider is at the position of closet approach to the fixed magnet, the glider KE is momentarily zero (we also call it equilibrium position).
• At equilibrium position, all of the energy of the system is stored in the magnetic field as magnetic potential energy, and this energy equals the gravitational force component on the glider parallel to the track.
• We measure the angle using our phone or the angle measurer in the lab (Figure 3)

Figure 3: Measuring the angle.

Approach.

• Because we don't know an equation of magnetic potential energy, we call magnetic potential energy U(r) as a function of separation distance r.
• We will calculate U(r) by integrating the force F(r) in term of r which is the separation distance between two magnets.
• To find the function of force in term of r, we will use Logger Pro. We thus will enter the data for force and r (after calculation and measurement), graph them, and obtain the equation.
• We predict that force F(r) and r is related to each other though this equation : F(r) = Arⁿ , while force is equal the gravitational force component on the glider parallel to the track.

Calculating force F and measure the separation distance r:

• We first draw a free body diagram

Figure 4: Drawing a free body diagram and figure out the force of magnet.

• Since we derive F=mgsin⦶, we then need to measure the angle and the mass of glider.
• Calculating force is shown below

Figure 5: Calculating the force.

• Measuring the separation distance r.
  
  Figure 6: Measuring the separation distance between two magnets.

Working with Logger Pro

1. We first enter the value of force and separation distance that we just got above into Logger Pro.

Figure 7: Data for force and distance.

1. We then curve fit the data and obtain the graph of force F vs. separation distance r.

Figure 8: Curve fit the graph and analyze the equation

       Now we have a function of force in term of separation distance r: F(r) = 0.0002038 * r^-1.871, we then integrate to get the magnetic potential energy. The work is shown below

Figure 9: Integrating F(r) to obtain an equation for U(r).

U(r) = 0.000234 * r^-0.871

Part 2: 

      Verify the theory of the conservation of energy applies to this experiment.

Approach:

• From part 1, we know that when the glider moves toward the magnet, all of kinetic energy is transferred to magnetic potential energy.
• To prove the conservation of energy, we need to find the initial kinetic energy and the final magnetic energy.
• Our prediction is that KE_intital = MPE_final or total energy = constant number.
• To find kinetic energy, we need mass and velocity, while to find magnetic energy we need the separation distance between two magnets.

Procedure:

1. We use the same set-up as part 1, however, we add a motion sensor behind the magnet to record the velocity of the glider.
2. With the air source turned off, place the glider on the air track, reasonably close to fixed magnet. Run the motion detector. Determine the relationship between the distance the motion detector reads and the separation distance between the magnets.
3. Using books to tilt the track at various angles, measure the angle, then give the glider a gently push so that it moves toward the magnet.
4. Set up the Logger Pro, create New Calculated Column that will let us get the separation between the magnets from the position as measured by the motion detector. 
5. Create New Calculated Columns to calculate the kinetic energy, magnetic potential energy, and total energy of the system as a function of time.

Data Collection

• Following the procedure above, we will use Logger Pro to collect the velocity of the glider. After giving the glider a push, hit the Collect button, and the data appears on Logger Pro like shown below.
  
  Figure 10: Data collecting for time, position, and velocity.

Data Analysis:

• We first calculate the initial distance 

Figure 11: Calculating the initial distance.

Initial distance = 0.3045m

• Calculating the separation distance r

Figure 12: Calculating separation distance.

• Calculating kinetic energy

Figure 13: Calculating kinetic energy.

• Calculating magnetic potential energy

Figure 14: Calculating magnetic potential energy.

• Calculating total energy

Figure 15: Calculating the total energy.

      When we're done calculation, we got the result as shown below

Figure 16: Result table after calculation.

     We then plot the graph of kinetic energy, magnetic potential energy, and total energy as a function of time in Logger Pro.

Figure 17: Graph of kinetic energy, magnetic potential energy, and total energy.

      What we expected was that the total energy was supposed to be a horizontal line in order to prove the theory of the conservation of energy. However, what we obtain is not actually a horizontal line, thus there are some possible errors happening during the experiment. These are:

• The angle we measure has an uncertainty (± 0.1) as our phone can't give us the exact number
• The separation distance also has an uncertainty (± 0.05) based on our ruler.
• When measuring the velocity of the glider using Logger Pro, we may press Collect button slower than the actual time the glider gets moving, thus the value we get for velocity may affect our kinetic energy.
• Since separation distance and the angle both have uncertainty, these lead to the uncertainty in magnetic potential energy because the equation is derived from F(r) while F is a function of ⦶ (F=mgsin⦶)
• We also assume that the air track is ideal frictionless; however, it is not perfectly smooth, thus this condition may affect our experiment.

Conclusion:

      In this lab, we perform experiments in order to obtain two goals:

1. To come up with the equation of magnetic potential energy.
2. To prove that the theory of the conservation of energy

      Our result for part 1 is : U(r) = 0.000234 * r^-0.871 

      For part 2, we're close to obtain the horizontal line for total energy. However, we explain reasons why we don't get a perfectly horizontal line. Actually, there are many errors that we hardly avoid which may affect our experiment. But overall, our result is "pretty good" to prove the theory of the conservation of energy in this experiment.

Conservation of energy-Mass-spring system

Objective:
    The goal of this lab is to prove the theory of conservation of energy which means no matter where the object is, the total energy has to be the same.
How the lab works out:
    This lab is actually divided into two parts:
       Part 1: Determine the value of spring constant.
       Part 2: Using the value of spring constant to calculate the elastic potential energy of the spring, then determine the total energy.
Part 1: Determining the value of spring constant.
Set-up:

Figure 1: The apparatus 

     1. Using a C-clamp to secure a vertical rod to the table, then mount a horizontal rod to the vertical rod.
     2. Put a force sensor on the horizontal sensor, and put a motion sensor on the floor facing up.
     3. Calibrating the force censor using zero mass and a 1 kg weight, then remove the weight.
     4. Attach a string to the force sensor, then zero the force sensor.
     5. Place a 100-gram the mass hanger on the spring, hold it so that the spring is un-stretched.
     6. Open a file called L11E2-2 (Stretching Spring) on Logger Pro, and under the motion sensor setup, select Reverse Direction and also zero the motion sensor with the mass hanger in this position.
     7. We then (let the spring pull down with 100-gram mass hanger on it) and click Collect.
The apparatus of the experiment is shown in figure 1
Data Collecting:
      From the previous lab, we know that force and position are related to each other, and the relationship is shown through the following equation

                      F = k * x                                                      

                                                                                             
   
      Thus, we need a graph of force vs. position, and the slope of the graph would give us the value of the spring constant.
      After hit the Collect button, below is what we get.

Figure 2: The graph of force vs. position

 Data Analysis:

      To be able to obtain the value of spring constant which is the slope of the graph, we first go to the Data: Sort Menu and Sort on position. Then we linearize the graph and obtain the result:

m = 8.487N*m

      Since the value of slope is the value of spring constant, we finish part 1 with k = 8.487 N*m.

Part 2: Proving the conservation of energy by showing that the total energy keeps constant at any position.

Set-up:

    1. We use the same set-up like part 1 except that in this part, we will add an additional 200-gram on the mass hanger.

    2. Record the position of the spring when it is un-stretched. (Position of the spring is relative to the floor, not to the horizontal rod)

    3. Let the spring pull down with the mass attached to it, then record the new position of spring relative to the floor. 

      The motion sensor we put on the floor in part 1 will help us figure out the position of the spring, and the result is shown in Logger Pro. To make it easy for the motion sensor to read, you can attach a piece of paper under the mass hanger. How the experiment performed is shown in figure 3.

Figure 3: The spring is stretched under the weight of 300-gram mass hanger. The motion sensor records the position of the spring relative to the floor.

Working out some preliminary stuff that physics 2AG ignores:

     Back to physics 2AG, we ignore the mass of spring and also the movement of the bottom end of the spring. However, in physics 4A, we consider the effects of these both factors when calculating the total energy of the whole system. While the mass of the spring itself contributes to the potential energy of the system, the movement of the bottom end of the spring contributes to the kinetic energy of the whole system, thus it's important to measure them both. 

     First, we would figure out the gravitational potential energy of the spring itself. The work to find out the equation is shown in figure 4.

Figure 4: Determining the gravitational potential energy of spring itself

     As a result, we obtain this equation to show for the gravitational potential energy of spring itself

GPE_spring = (0.5) * m_spring * g * y_o 

     Then, we work out to find the kinetic energy of spring itself. The work is shown in figure 5.

Figure 5: Figuring out the kinetic energy of spring itself.

     The result we obtain to calculate kinetic energy of spring itself is the following equation:

KE_spring = (0.5) * (m_spring / 3) * (v_end²)

     We examine that as the spring is stretched, it also has the elastic potential energy, thus we need an equation to calculate elastic potential energy. 

Elastic PE = (0.5) * k * (∆y)²  

     As we look at the whole system, we will have all these equations below in order to calculate the total energy. 

Figure 6: Equations to calculate KE(system), GPE(system), and Elastic PE

Data Analysis:

    Since we have all equations we need, we make a new Calculated Column in Logger Pro, enter equations and let Logger Pro calculate the data for us. Before entering the equations into Logger Pro, we also measure the mass of spring itself, determine the value of un-stretched spring, and the weight of mass hanger.

m_spring = 87g = 0.087kg

m_hanging = 300g = 0.3kg

Un-stretched position = 0.662m

    Now, we start entering equations into Logger Pro. First, calculating the kinetic energy of hanging mass.

Figure 7: Setting up equation in Logger Pro to calculate the kinetic energy of hanging mass.

     Then, calculating the kinetic energy of spring itself

Figure 8: Equation to calculate kinetic energy of spring itself in Logger Pro

     Calculating gravitational potential energy of spring itself.

Figure 9: Equation to calculate gravitational potential energy of spring itself in Logger Pro.

Calculating gravitational potential energy of the hanging mass.

Figure 10: Equation to calculate gravitational potential energy of hanging mass in Logger Pro.

Calculating elastic potential energy of spring.

Figure 11: Equation to calculate elastic potential energy of spring in Logger Pro.

    The last step is that we calculate the total energy.

Total energy = GPE_hanging + GPE_spring + KE_spring + KE_hanging + Elastic PE

    When we're done the calculation, below is our result

Figure 12: The final result: total energy besides GPE_hanging , GPE_spring , KE_spring , KE_hanging , and Elastic PE

 We then graph all our results to check the pattern of each graph.

Figure 13: Graph of GPE_hanging , GPE_spring , KE_spring , KE_hanging , and Elastic PE

    Finally, we graph the total energy. What we want to obtain is a horizontal line which means that the total energy keeps constant no matter where the position of the spring is. Below is our result.

Figure 14: Graph of total energy.

      

      We notice that the graph of total energy is not a horizontal line like what we expected, thus we may have some errors while performing the experiment. 

      Some possible errors may be: 

           1. When we determine the value of spring constant in part 1, the graph of force vs. position is not an actual straight line, thus we may obtain a value of spring constant off a little bit from actual value, which affects the elastic potential energy in part 2.

           2. There are some human errors in collecting the force of hanging mass on the spring because we may hit the Collect button a little bit faster or slower than the actual time to let the hanging mass pull the spring down.

           3. When we zero the motion sensor and force sensor, but we didn't get 0 for both of them on Logger Pro.

 Conclusion:

      In this lab, we try to prove the theory of the conservation of energy, which means that the total energy keeps constant wherever the position of spring is. We divide the lab into two parts. In part 1, we determine the value of spring constant k = 2.861N/m. Then we use that value to calculate elastic potential energy of spring in part 2 together with GPE_hanging , GPE_spring , KE_spring , KE_hanging . From that, we can calculate the total energy and graph to check whether it is a horizontal line or not. The graph we obtain is not actual a horizontal line, but we explain some possible errors along the way we perform the experiment. Although the result is not exactly like what we expect, overall we reach our goal is to  show that the total energy is "theoretically" constant.

    

     

    

   


                                                         
                                                               
                                                                                                                         

Work-Kinetic Energy Theorem Activity

Objective: 
      - Figuring out the spring constant from force vs. position graph
      - Determining the relationship between the work done and kinetic energy.
      - Verifying that the work done is equal the kinetic energy
Part 1:  Work Done by a Non-constant Spring Force.
    Objective: 
            Figuring out the spring constant of our spring and use the integration routine in the software to find the work done in stretching the spring.
    Equipment needed: 
            Ramp, cart, motion detector, force probe, spring, weight, C-clamp, and computer.
Set-Up: 
    1. Let the ramp lie on the horizontal table.
    2. Using the C-clamp to secure the force probe at the edge of the table.
    3. Let the motion detector sit on the other side of the ramp, and connect it with the Logger Pro.
    4. Connecting the spring to the force probe.
    5. Connecting the other side of spring to the cart whose body holds some weight and a square.
Figure 1 shows the apparatus of the experiment.

Figure 1: The apparatus of the experiment

How to start the experiment:
    1. Calibrating the force probe with a force of 4.9N applied.
    2. Zero the force probe and the motion detector with the spring unstressed or uncompressed.
    3. Set the motion detector to "Reverse Direction" so that when the cart moves toward the motion detector, we would obtain the positive direction.
    4. Then slowly push the cart move slowly towards the motion detector until the spring is stretched about 1.0m.

Data Collecting:
    Since we connect the motion detector with the Logger Pro, we shall get a graph of force vs. position as the cart moves toward the motion detector.
    Below is our graph force vs. position (Figure 2)

Figure 2: The graph of force vs. position graph

    Back in physics 2AG, we know that as the spring stretches, force is related to position through an equation:

F = k * x

 F: force applied to the cart

k: constant spring

x: the stretch of the string.

    Since the relationship between force and position of spring is linear, we can figure out the value of the constant spring by taking the value of slope of the graph force vs. position. Thus we linearize the graph of force vs. position and obtain the value of slope like below.

Figure 3: Linearizing the graph of force vs. position graph and obtaining the value of slope.

    In this graph, we obtain the value of m = 2.855N/m, thus the value of spring constant is 2.855N/m.

    After figuring out the value of spring constant, we use the integral to calculate the work done in stretching the spring. The calculation is shown in figure 4.

Figure 4: Integrating the area to find the work done.

W = 0.1541 J

Part 2: Kinetic Energy and The Work-Kinetic Energy Principle

Objective: 

       Determining the relationship between the work done and the kinetic energy.

Set-up:

       We use the same set-up in part 1, which is shown in figure 1.

How to perform the experiment:

   1. In this part, we first measure the mass of the cart. (Figure 5)

Figure 5: Measuring the mass of the cart.

m(cart) = 574g.

    2. We set up a new column in Logger Pro to calculate the kinetic energy which is this equation:

KE = (1/2) * m * v²

    ^{3. Zero the force probe and the motion detector, then pull the cart so that the spring is stretched about 1.0m from the un-stretched position. (Figure 6)}

Figure 6: Pull the cart so that the spring is stretched around 1.0m

     4. Release the cart so that it moves toward the spring. 

^{Data Analts:}

    ^{As the cart moves toward the spring, we would obtain the graph of force vs. position using Logger Pro. Also, on the y-axis, adding the kinetic energy. The graph of force vs. position and kinetic energy vs. position are shown in figure 7.}

Figure 7: The graph of force vs. position (Purple) and kinetic energy vs. position (Dark Blue).

     Now to figure out the relationship between work done and kinetic energy, we use integral to calculate the work done and also examine the value of kinetic energy, then compare them. Below are three spots we compare work done and kinetic energy.

Figure 8: First position comparing work done and kinetic energy.

| W | = 0.1458N/m,

KE = 0.146N/m

Figure 9: Second position comparing the work done and the kinetic energy

| W | = 0.1232N/m,

KE = 0.123N/m

Figure 10: Third position comparing the work done and kinetic energy

| W | = 0.1076N/m,

KE = 0.107N/m

     From three different position comparing the work done and kinetic energy, we notice that the value of the work done and kinetic energy is almost the same. Thus, we can conclude that the work done is equal the kinetic energy

W = KE

Part 3: Work-KE Theorem

Objective: 

     Comparing the result of the work done and the kinetic energy in order to verify the relationship between two of them one more time.

Set-Up:

     In this part, we don't set up any equipment. The professor would turn on a video capturing the moment when a professor did an experiment to figure out the relationship between the work done and kinetic energy in the old time. In the video, the professor uses machine to pull back on a large rubber band, and the force exerted on the rubber band is recorded by an analog force transducer onto a graph. (Figure 11)

Figure 11: Professor is performing the experiment.

    The stretched rubber band is attached to a cart of known mass. Then the professor releases the cart to let it pass through two photo gates a given distance apart. Since we know the distance and time that the cart passes through the first and second photo gate, we can use them to figure out the speed in order to calculate the kinetic energy.

Figure 12: The set-up will give us the time and distance as the car passes through two photo-gates.

Data Collecting:

    As the cart passes through two photo gates, the professor records the time, the distance, and mass of the cart. 

m = 4.3kg

x = 15cm = 0.15m

t = 0.045s

Data Analysis:

    Since we have the mass of the cart, the time and distance, we can use this below equation to find out the kinetic energy

KE = (1/2) * m * v² 

    Calculating the kinetic energy:

Figure 13: Calculating the kinetic energy of the object.

KE = 23.89J 

    For the work done, we would stop the movie and make a careful stretch the force vs. position graph, then divide the graph into calculable shapes that allow us to use equation to calculate the work done such as triangle, trapezoid, or rectangular. 

    Calculating the work done:

Figure 14: Calculating the work done.

W = 26.6J

    We have W = 26.6J and KE = 23.9J. However, we found out in part 2 that the work done "have to be" equal the kinetic energy of the object. This means that there are some errors while performing this experiment.
    Some possible errors in this experiment leading to the unequal value of work done and KE.
        1. As professor pull back the machine, she created three different graph of force vs. stretch. Because we randomly use one of three different graph to calculate the work done, we will get the result that a little bit off from the actual value.
        2. While we calculate the work done, we also estimate the height or base of triangular or rectangular. Therefore, the result we get for work done have some uncertainty with it.
        3. The equipment that professor used in this experiment are too old that may not give us the accurate value like our equipment today. What I mean right here is the photo gate and the time watch that professor used in this experiment.
Conclusion:
      In this lab, we manage to find the spring constant in part 1. The result is k = 2.855N/m. In part 2, we determine the relationship between the work done and kinetic energy of the object. After checking three different positions on graph force vs. position, and kinetic energy vs. position, we find out that the work done is equal the kinetic energy. In the last part, we will use our result in part 2 to check an old experiment. We calculate the work done and kinetic energy of the old experiment: W = 26.6J and KE = 23.9J. The results of work done and kinetic energy should be the same, though they are actually not. Therefore, we can conclude that there were some possible errors during this experiment. We come up with three possible errors that explained in part 3.

   

   

   

Centripetal force with a motor

Objective:

1. Find the relationship between the angular speed and the angle formed by the apparatus
2. Using equation found in part 1 to calculate several theoretical values of angular speed responding to different angles.
3. Measure the period and use it with other known condition to find the experimental value of angular speed.
4. Compare experimental values and theoretical values of angular speed.

Set-up:
    How to set up the apparatus of the experiment:
        1. Mounting an electric motor on a surveying tripod.
        2. A long shaft going vertically up from the shaft.
        3. Mounting a horizontal rod on the vertical rod
        4. A long string is tied at the end of the horizontal rod.
        5. A rubber stopper at the end of the string.
        6. A ring stand with a horizontal piece of paper sticking out.

Figure 1: The apparatus of the experiment.

How the experiment is performed:
       1. Turn on the electric motor to let the string spin.
        2. As the motor spins at a certain angular speed w, the mass revolves around the central shaft at a certain radius and the angle.
        3. Performing 6 trials with an increasing amount of voltage each time and collecting the data.
        4. The data needs collected each trial is the time that the motor need to travel in a certain number of revolutions and the height of the horizontal paper attached to the ring stand as the motor grazes it when the motor spins.
     The diagram is shown below help you imagine easier what we are going to measure and collect.

Figure 2: The diagram shows how equipments are set up in this experiment.

Data Collection:

• Professor performs the first trial to show us how the system works, he then lets us measure necessary quantities such as length of the string, height of tripod, etc...to calculate the angular speed and check whether it makes sense.
  
  Figure 3: Professor is performing the first "checking" trail.
• If the result we get make sense, we will start doing real trials with increasing speed of motor, results in a wider angle, and higher height which is measured by the ring stand.
  
  Figure 4: Performing real trials with different speed of motor, different angle and height.
• We will do 6 trials and record appropriate data.

Data Analysis:
      Since our objective is to find out the relationship between the angle and omega (⦶ and w), we need to come up with an equation to solve for angle and omega.
      As we look at the diagram, we notice that the string make with the height of the tripod an angle that we can find thanks to the right triangle. The height is equal the difference of the height of the surveying tripod and the height of a horizontal piece of paper which the rubber stopper hits.
      After finding an equation to solve for the angle, we come up with another equation to solve for omega as a function of the angle and measured data (height and length of string).
      Below is the derivation of an equation for the angle and omega.

Figure 3: Deriving an equation to solve for the angle and omega.

      

        Since we have both equation, what we need is data to plug in. We performed six trials in this experiment with different angle and period at each time. For each time, we increase the power of the electric motor, the angular speed (w) increase, the mass revolves around the central shaft at a larger radius and the angle also increase. Figure 4 is our data.

Figure 4: Height and time are recorded for each trial in order to find out the angle and angular speed.

      We then plug our data into our equation to calculate the angle and the angular speed. Below is an  example how we calculated them. (Figure 5)

Figure 5: Calculating the angle and the theoretical value of angular speed.

      We keep doing the same process for the next 5 trials, and here are our results.

Figure 6: Results for the angle and angular speed after calculating all six trials.

       One of the data we also recorded in this experiment is the time that is required to finish one revolution for each trial. With that data, we are able to find the experimental value of angular speed through an equation:

w = 2∏ / T

      Example of calculating the experimental value of angular speed.

Figure 7: Calculating the experimental value of angular speed based on the time we recorded.

       

       We perform the same process for the next five trials. After we're done, we put them into the same table with the angle and theoretical value of angular speed. The table is shown in figure 8.

Figure 8: The results of experimental value of angular speed along with these previous results.

      Since we have the experimental and theoretical value of angular speed, we enter them into the Logger Pro and graph them to check how well we did with this experiment. If we obtain the slope of the graph equal 1, this means that our experimental values match with our theoretical values. In case the slope is not exactly equal 1, this refers that we did something wrong in the experiment or made wrong assumption.

     So we go and enter the data into the Logger Pro.

Figure 9: Entering experimental and theoretical value of angular speed into Logger Pro.

      Then we linearize the data and look for the value of the slope of the graph.

Figure 10: Graph showing the relationship between experimental and theoretical value of angular speed.

m = 1.005 

      As we get the value of the slope m = 1.005, we can conclude that our experimental values almost perfectly match with the theoretical value of angular speed. This also means that we follow the procedure exactly and perform a great experiment.

Conclusion:

    In this experiment, our objective is to find the relationship between the angle and the angular speed. We found that angular speed is related to the angle through an equation in which the angle is calculated thanks to the right triangle with the hypotenuse L and height H-h.

     The slope of the graph between experimental and theoretical value of angular speed is:

m = 1.005

     Although the slope is not exactly equal 1 like our expectation, the result is not far off from 1, which indicates that our results are pretty good.

     The slope is not exactly equal 1 because of some of the following reasons.

         1. We couldn't record exactly the time for the the string finished one revolution, thus the error may interfere with our experimental value of angular speed.

         2. We may make unnecessary rounding number when calculating the theoretical value of angular speed. 

         3. The uncertainty in measuring the height of the surveying tripod, the ring stand with the horizontal paper, and the length of the string also affect our results.