Conservation of Energy in a Spring based Launcher

Transcription

1 Conservation of Energy in a Spring based Launcher Jenna Pralat Lab Partners: Melissa Farraher, Nathan Ng, Rithika Senthilkuar Section A Deceber 9, 016 Introduction Purpose: The purpose of this lab was to analyze the conservation of energy in a spring based syste. Researchable Question: How does increasing the copression of a spring, which launches a projectile at a 40 angle, affect the range of the projectile. Hypothesis: Increasing the copression of a spring, will increase the range of the projectile, where d T [Δx] = Δx Δx

2 Methodology Procedure: Finding k To find k, the launcher was placed on its side, with the launching echanis pointing up. A eter stick was placed at the base of the arble launcher in order to easure the Δx of the spring. A string was tied to the handle on the spring and a force eter. The force eter was pulled a specific distance fro its starting position and the force recorded. Procedure: Measuring Range Align the initial x position of the arble with the start of a easuring tape, and the body of the launcher with the line between the tiles on the floor, the easuring tape is also along this line. Pull back the handle on the launcher to the slot for the given trial. After the person launching the arble counts down 3,, 1, go the handle is pushed parallel to the slot, releasing the spring which will launch the arble. This first run is used to infor the group the area in which the arble will land. In the following trials, on go the arble is launched and the position where the arble lands is recorded. For the first three slots on the launcher the final position was arked by a finger and easured with the tape easure right away. For the fourth and fifth setting the position was arked with tape and easured afterwards.

3 Methodology cont. Δx v i θ y i a Ty v f Constants = 6.13 g = kg yi = 4.7 c = 0.47 yf = 0.00 aty = -9.8 /s θ = 40.0 Figure 1: Diagra of the apparatus with key variables. Equations d t v it = Δx h = 0 t T = Δx Δx d T = Δx Δx v i = 1.174(.839d f.47) y f

4 Figure : Photo of our group s set up for data collection. The gray barrel is where the spring is located.

5 Force, F (N) Results: Calculating k F = 64.8x R² = Copression of Spring, Δx () Figure : This graph shows the Force vs Copression of the spring and the line of best fit for the data. The slope of the line represetents the k value of the spring. For this spring the k vaule is 64.8 N/. Force Δx (N) () k 64.8 Table 1: Forces and copressions for data in the graph.

6 Results: Data Suary Δx davg STDEV %RSD dt %err () () () of davg () of d Avg 1.77 Avg 34.3 Table : Suary of data for the range of the spring at the five IV settings. TEi TEf (J) (J) %E Change Table 3: Initial Total Energy (TEi) and Final Total Energy (TEf)

7 Range, d () Results: Graph d = 105.8x x R² = d = x x R² = Copression of Spring, Δx () Theoretical Range Measured Range Figure 3: Graph of range of projectile vs copression of the spring. With trend lines and equations for those lines. The top equation is for the theoretical Range and the botto equation is for easured range, both in ters of copression of the spring.

8 Analysis The average range for the ten trials was used to create the easured range points on the graph in Figure 3. The precision of the data was high with a %RSD being However, accuracy was low with a %error of The atheatical odel for easured data is strong with a R value of.99. The doain and range for the graph are restricted to the first quadrant. The range is restricted because the range of the projectile cannot be negative. The doain is restricted because with our apparatus the spring could not ove in the other direction. The x and y-intercepts of the graph should be 0 because if there is no copression of the spring, launching the arble, then the arble will not be able to travel any distance. The total energy in the beginning is the su of the potential energy of the spring and the potential energy of gravity. At the end the total energy is now just kinetic energy. The total energy final is less than the total energy initial. This error is because of the fors of energy that were not accounted for. Work lost to friction, sound energy, and theral energy. If these three sources of energy were considered the Total Energy final would be greater than it was calculated to be.

9 Conclusion The data fro the experient supports the original hypothesis that as the copression of the spring increases the range of the projectile increase. As the aount the spring was copressed increased so does the range of the projectile. The total energy final was less than the total energy initial because sources of energy were left out. These sources are theral, sound, and friction. Extensions on this lab could be changing the angle the spring launches at or easuring the distance the arble rolls after landing to understand the non-conservative forces at work.

10 Appendix 1: Equation Derivations Theoretical Initial Velocity in ters of Theoretical Tie in ters of copression of spring copression of spring PE s + PE g = KE i + PE g 1 kδx + gy i = 1 v it + gy i 1 kδx = 1 v it v it = kδx v it = Δx v ixt = cosθ kδx t T = t T = y f = 1 a Tyt T + v iyt t T + y i 0 = 1 a Tyt T + v iy t T + y i v iy v iy 4 ( 1 a Ty) (y i ) t T = ( 1 a Ty) kδx sin θ ( kδx sin θ) a Ty y i a Ty kδx sin θ kδx sin θ a Ty y i t T = a 64.8 Δx sin Δx sin ( 9.8)(.47) ( 9.8) t T = Δx 4369 Δx

11 Appendix 1: Equation Derivations Theoretical Range in ters of copression of spring d T = v ixt t T d T = cosθ kδx kδx sin θ kδx sin θ a Ty y i a 64.8Δx d T = cos Δx sin Δx sin ( 9.8)(.47) d T = Δx Δx Initial Velocity in ters of tie d f = 1 (0)t T + v ix t T + 0 d f = v i cosθt T t T = d f v i cosθ y f = 1 a Ty t T + v iy t T + y i 0 = 1 a Ty ( d f v i cosθ ) d f + v i sinθ( v i cosθ ) + y i 0 = a Ty d f v i cos θ + tanθ d f + y i tanθ d f y i = a Ty d f v i cos θ v i ( tanθ d f y i ) = a Ty d f cos θ v a Ty d f i = cos θ( tanθ d f y i ) a Ty d f v i = cos θ( tanθ d f y i ) v i = cos 40( tan40 d f.47) v i = 1.174(.839d f.47)

12 Velocity in x direction v ix = cosθ 1.174(.839d f.47) v ix = cos (.839d f.47) v ix = (.839d f.47) v ix = (.839d f.47) Appendix 1: Equation Derivations Velocity in y direction v iy = sinθ 1.174(.839d f.47) v iy = sin (.839d f.47) v iy = (.839d f.47) v fy = v iy + a Ty Δy v fy = v iy + a Ty (y f y i ) v fy = ( (.839d f.47) ) + a Ty (0 y i ) Velocity Final v f = v x + v fy v fy = ( (.839d f.47) ) + ( 9.8)(.47) v f = ( (.839d f.47) ) + ( ( (.839d f.47) ) + ( 9.8)(.47))

13 Appendix : Full Data Tables Δx d 1 d d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 10 d avg STDEV %RSD d T %err () () () () () () () () () () () () () of d avg () of d Avg 1.77 Avg 34.3 v xt v it v iyt t T v i v x v y i v y f v f TEi TEf (/s) (/s) (/s) (s) (/s) (/s) (/s) (/s) (/s) (J) (J) %E Change