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.