Lab 06 Force and Motion Part II

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University of Cincinnati, Main Campus *

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2001L

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Mechanical Engineering

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Apr 3, 2024

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docx

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Lab 06: Force and Motion Part II I. Develop an experimental mathematical model to describe the behavior of a system a. Select an IV to test Hanging mass b. Experimental Design Template Research Question: How does the acceleration of a system change when hanging mass changes? Dependent variable (DV): Acceleration of a system Independent variable (IV): Hanging mass Control variables (CV): Mass of the system: 0.3044kg, length of the string: 1.04m, starting point: 0.85m Testable Hypothesis: There is a positive correlation between the hanging mass and the acceleration of a system. Prediction:
Picture 1: Experimental setup c. Complete the experimental design. We will be doing 8 trials and will use values of 0.0094kg, 0.0144kg, 0.0194kg, 0.0244kg d. Conduct the experiment. The uncertainty of the measured values for acceleration is ±0.001m/s2 due to the rotary motion sensor’s estimated scale uncertainty, given in this lab. The uncertainty of the measured values for the length of the string and position of the system is ±0.0005m due to the meter stick’s estimated uncertainty, given in a previous lab. The uncertainty of the masses is ±0.001kg due to the scale’s estimated uncertainty, given in a previous lab. Hanging mass (kg) Trial 1 (m/s 2 ) Trial 2 (m/s 2 ) Trial 3 (m/s 2 ) Average (m/s 2 ) 0.0094 0.288 ± 5.2*10 -4 0.289 ± 2.8*10 -4 0.290 ± 2.2*10 -4 0.289 0.0114 0.349 ± 3.0*10 -4 0.346 ± 6.2*10 -4 0.349 ± 5.1*10 -4 0.348 0.0134 0.407 ± 3.4*10 -4 0.407 ± 4.2*10 -4 0.404 ± 1.1*10 -3 0.406 0.0154 0.467 ± 5.0*10 -4 0.467 ± 5.3*10 -4 0.466 ± 4.9*10 -4 0.467 0.0174 0.525 ± 2.8*10 -4 0.526 ± 6*10 -4 0.524 ± 6.4*10 -4 0.525 0.0194 0.575 ± 7.1*10 -4 0.574 ± 5.7*10 -4 0.575 ± 5.5*10 -4 0.575 0.0214 0.630 ± 5.3*10 -4 0.630 ± 6.5*10 -4 0.630 ± 6.9*10 -4 0.630 0.0234 0.683 ± 6.8*10 -4 0.684 ± 9.4*10 -4 0.677 ± 2.8*10 -3 0.681 Data table 1: Acceleration vs Hanging mass e. Enter collected data into Excel 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 f(x) = 0.29 x + 0.03 R² = 1 Hanging Force vs Acceleration Hanging Force (N) Acceleration (m/s^2)
Graph 1: Acceleration vs Hanging force (The error bar for acceleration and hanging force is too small to be seen) f. Consider the mathematical model provided by Excel a = 2.8663F + 0.0295 2.8663 (1/kg) and 0.0295 (m/s 2 ) A causal relationship exists between the acceleration (a) and the gravitational force (F) if the mass of the system, length of the string, starting point is held constant, indicating a positive linear function. II. Developing a second experimental mathematical model to describe the behavior of the system. a. Select a second IV. Mass of the system b. Repeat all steps in Part I. Experimental Design Template Research Question: How does the acceleration of a system change when the mass of the system changes? Dependent variable (DV): Acceleration of a system Independent variable (IV): Mass of the system Control variables (CV): Hanging mass: 0.0234kg, length of the string: 1.04m, starting point: 0.85m Testable Hypothesis: There is a negative correlation between the mass of the system and the acceleration of a system. Prediction: c. Complete the experimental design. 8 trials, values of 0.3044kg, 0.3544kg, 0.4044kg, 0.4544kg, 0.5044kg, 0.5544kg, 0.6044kg, 0.6544kg d. Conduct the experiment. The uncertainty of the measured values for acceleration is ±0.001m/s2 due to the rotary motion sensor’s estimated scale uncertainty, given in this lab.
The uncertainty of the measured values for the length of the string and position of the system is ±0.0005m due to the meter stick’s estimated uncertainty, given in a previous lab. The uncertainty of the masses is ±0.001kg due to the scale’s estimated uncertainty, given in a previous lab. System mass (kg) Trial 1 (m/s 2 ) Trial 2 (m/s 2 ) Trial 3 (m/s 2 ) Average (m/s 2 ) 0.3044 0.687 ± 4.8*10 -4 0.688 ± 5.6*10 -4 0.687 ± 1.3*10 -4 0.687 0.3544 0.596 ± 9.5*10 -4 0.595 ± 1.0*10 -4 0.597 ± 3.4*10 -4 0.595 0.4044 0.528 ± 6.7*10 -4 0.528 ± 5.2*10 -4 0.528 ± 5.6*10 -4 0.528 0.4544 0.473 ± 4.4*10 -4 0.472 ± 4.8*10 -4 0.471 ± 8.1*10 -4 0.472 0.5044 0.428 ± 4.6*10 -4 0.429 ± 4.1*10 -4 0.428 ± 4.7*10 -4 0.428 0.5544 0.391 ± 4.1*10 -4 0.392 ± 2.2*10 -4 0.392 ± 5.8*10 -4 0.392 0.6044 0.361 ± 4.0*10 -4 0.362 ± 4.0*10 -4 0.361 ± 3.3*10 -4 0.361 0.6544 0.335 ± 2.5*10 -4 0.335 ± 3.9*10 -4 0.335 ± 3.1*10 -4 0.335 Data table 2: Acceleration vs System mass e. Enter collected data into Excel 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 f(x) = 0.23 x^-0.94 R² = 1 System Mass vs Acceleration System Mass (kg) Acceleration (m/s^2) Graph 2: Acceleration vs System mass (The error bar for acceleration and system mass are too small to be seen)
f. Consider the mathematical model provided by Excel a = 0.2252 m 0.938 0.2252 (N) and -0.938 A causal relationship exists between the acceleration (a) and the system mass (F) if the hanging mass, length of the string, starting point is held constant, indicating a negative power function. III. Connecting Experimental Model to Established Scientific Model a. Define established scientific model for the acceleration of a system. a ∝ F net , hanging mass vs. acceleration of a system a 1 m , mass of a system vs. acceleration of a system b. Compare your experimental model with the established scientific model. a = 2.8663 F + 0.0295 , acceleration of system a = 0.2252 m 0.938 , mass of system vs acceleration of system The relationships represented in our mathematical models and the scientific equation are remarkably similar to the established models for the acceleration of a system in relation to the hanging mass, as the hanging mass and the acceleration of a system are proportional, while the mass of the system is inversely proportional to the acceleration of the system. The only difference in the models is that the mass value in the equation for the mass of a system vs. acceleration of a system has an exponential value of 0.938, which may be partially attributed to the uncertainty values of the data.
IV. Newton’s Second Law a. Connect your experimental models to the general form of Newton’s Second Law as Σ F = m system a , which can be rewritten for motion in one dimension as: a x = Σ F m system = F 1 x m system + F 2 x m system + F 3 x m system + i. Compare your group’s mathematical model, which has the form a = C 1 F app + C 2 to the general form of Newton’s Second Law. In the section of your lab records that includes this model, indicate what the constants C 1 and C 2 physically represent. Then, determine what the value for C 1 should be based on your lab set-up and compare it to the value in your model. Knowing that your model describes the behavior of a real system, summarize the conditions of the lab set-up that might cause the values for C 1 and C 2 to be larger or smaller than expected. Our group’s mathematical model is a = 2.8663F + 0.0295, with C 1 =2.8663 and C 2 =0.0295. This relates to the general form of Newton’s Second Law because C 1 represent 1 m , and since F is multiplied by C 1 , it is the same as Newton’s second law. C 2 is the acceleration. Hanging mass Acceleration Force of gravity on system System Mass Fnet on cart in direction of cart's motion (a)Accelerati on of System Experimental relationship: a=2.8663F+0.0295 Fnet increases with larger mhanging Fnet increases with larger acceleration Fnet increases with larger Fg No impact (controlle d) Experimental relationship: a=0.2252/(m^(0.938)) Established relationship: a ∝1/m (when F is constant) Mechanism explained: There is no mechanism to describe here Established relationship: a Fnet (when msystem constant Mechanism explained: There is no mechanism to describe here
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