CONSERVATION OF ENERGY LAB. By: John Ta Lab Partners: Charan & Ram

Transcription

1 CONSERVATION OF ENERGY LAB By: John Ta Lab Partners: Charan & Ram

2 Introduction The purpose of the lab was to design and perform an experiment which analyzes the conservation of energy in a spring-based system. The researchable question was, How does increasing the distance at which a pinched rubber band is pulled back on its perpendicular bisector affect the distance that a frictionless cart at rest travels up a ramp? The hypothesis was, If the distance at which a pinched rubber band is pulled back at on its perpendicular bisector increases, then the distance the frictionless cart will travel will also increase (Δx d avg ).

3 Materials Ramp Two chairs Duct tape Masking tape Two rubber bands Meter sticks Vernier Force Sensor Logger Pro Excel PowerPoint Word Seven physics textbooks

4 Diagrams C! d avg! PE gc! Cart! L 0! a! v! Physics! Physics! Physics! Physics! Physics! B! L T! PE gb! +!KE B! "x! d avg!!! Chair! A! Cart!!x! PE ga!+!pe sa!

5 Procedure Two rubber bands were first taken and tied in the middle to form one rubber band shaped like the number eight (8). A preliminary test was then performed using a Vernier force sensor and Logger Pro computer software to determine the spring constant of the rubber band before the experiment. To do this, Ram held the sensor down and John pulled the rubber band to 30, 40, 50, 60, and 70 cm lengths marked on the meter stick. The slack length of the rubber band was then found. Two chairs were placed a random distance apart, and one end of the ramp was then placed on the seat of one chair and the other end of the ramp was placed underneath the other chair. Both chairs had their back legs pointing towards one another. Three legs of the chair with the ramp underneath it, and the top end and the bottom end of the ramp were duct taped to the floor or to the chair. At the 100 cm mark on the ramp, the height from that point was measured and was recorded. However, the hypotenuse length at that point was marked as 101 cm because there was a slight edge on the ramp that gave it a slightly longer length. The legs of the chair were inserted into the holes of the rubber band shaped like an eight (8). Then, the chair was shifted so the rubber band was directly above the 40 cm mark on the ramp. The rubber band s height was then calibrated by shifting it up and down in order to cover the entire back of the cart when pulling it back. After calibration, seven physics textbooks were placed on the chair with the rubber band to help prevent the chair from sliding back as the rubber band was pulled back.

6 Procedure (cont.) The cart was then placed on the ramp. John then pulled back the cart and rubber band five different lengths for each setting: 10, 15, 20, 25, and 30 cm back. Masking tape was used to mark the lengths of these points from 40 cm back. The cart was pulled back to one of the settings and once Ram said Go, the cart was then released and Ram took video of the cart using his phone as it traveled up the ramp. Five separate clips of video were created, each containing ten trials at each of the five settings. Afterwards, John and Ram analyzed the video in slow motion to find maximum distance that the cart had traveled up the ramp ten times for each length that the cart was pulled back. This data was recorded in Excel. After this, the entire setup was moved and recalibrated near the table at which Mr. Ellis generally sits at while he presents during class. Mr. Ellis then helped Ram and John pull back the Vernier Force Sensor two cm intervals down the ramp, starting at the original 40 cm starting mark, to measure the force that was returned at that specific displacement. To do this, Mr. Ellis would shout Go, and John would click Collect Data on Logger Pro for two seconds. Afterwards, John would take the average force over that two second timespan and would then shout to Ram the value, who would record that in Excel. The mass of the cart was measured and found to be kg. The post spring constant value was not found because of the lack of time.

7 Pictures

8 Constants & Equations Constants Slack Length of Rubber Band = L 0 = m Taut Length of Rubber Band on Chair = L T = m Starting position of the Rubber Band relative to ramp = P S = 0.4 m Spring constant = k = kg / s 2 Gravity = g = ±9.8 m/s 2 Angle = θ = degrees = radians Mass of cart = m = kg Equations PE $ = mgh F * = k x PE * = F x dx 1 2 1

9 Data Summary Δx d avg STDEV %RSD d theoretical %Error of (m) (m) (m) of d avg (m) of d avg IV IV IV IV IV Avg Avg

10 Graph Graph of d avg vs Δx y = x 2-1x + 4E- 15 R² = 1 d avg (m) d= x x R² = Δx (m) Experimental Theoretical

11 Analysis The line of best fit in the graph of the in of the independent value, Δx, vs the dependent value, d avg, shows a positive correlation, indicating that as the distance at which the pinched rubber band (Δx ) was pulled back at on its perpendicular bisector increased, the distance it traveled (d avg ) increased as well. This indicates that the group s proportionality statement, Δx d avg, is true for this experiment. Using the %RSD formula, it was determined that there was high precision in the results because the %RSD was 3.854%, which is less than 5%. Using the %Error formula, it was determined that there was low accuracy in the results because the %Error was greater than 10% (38.789%). The R 2 value in the graph, , of the independent value, Δx, vs the dependent value, d avg, shows that the line of best fit is strong, or has an R 2 value of greater than or equal to 0.95.

12 Conclusion What the group determined from the experiment was in line with what had been predicted with the hypothesis. As the distance at which the pinched rubber band was pulled back on its perpendicular bisector (Δx) increased, the distance it traveled up the ramp (d avg ) also increased. However, for IVs 1,2,3, and 5, the total energy at point A was less than the total energy at point C. This problem will be addressed later. A source of error that could have arose was the ways in which the data was collected. It is likely that the the total energies for IVs 1,2,3, and 5 at point A were less than point C because of this. Because of the high %Error, it is almost certain that there was problems with measuring accuracy of the data. For example, when the entire setup was re-setup and recalibrated, it is possible that the data was slightly off, and as a result of the small and precise numbers, loss of accuracy could have occurred. Another source of error that could have arose was ignoring the force of friction on the ramp. While it was assumed that the ramp was frictionless, it is impossible to achieve a frictionless surface in real life, and it is likely that the slight work of friction caused loss of energy. The group found the measured values to be as expected. As the pinched rubber band was stretched back further and further, the distance the cart traveled up the ramp increased more and more. As a follow up to this lab, adding friction to the surface that the cart would travel on would make things more complicated and interesting. This could possibly challenge the hypothesis.

13 Appendix A: Finding Angle of the Ramp θ = sin 78 ( ) where 23 centimeters is the height of the ramp at 101 a hypotenuse centimeters.

14 Appendix B: Derivation of PE sa Force of Spring equation: F x = k x L T! # - 2! Substitution of terms, A A + x A C ( D@ E 1 E F 1 E) is the componentized quantity for Δx, and where 2k represents the k of the pinched rubber band: F x = 2k C A A + x A C ( Simplification: F x = 4k( x) D@ E 1 E F 1 E) Integrating to find PE s at point A: PE *H = 4k Expanding the integral: PE *H = 4k 8 A x I A 8 A xa Simplifying: PE *H = 2k x I A x!x!!" # $ 2 & ' + ( +) '!

15 Appendix C: Energy Expressions at Point A & C K Energy at A:PE *H + PE $H Substitute: Energy at A: 2k x A I + P * x I sin θ mg Substitute in numerical values: Energy at A: x A I x I sin Simplify: Energy at A: x A I x I K Energy at C:PE $[ Substitute: Energy at C:mgh \ Substitute again: Energy at C: mg d ]^$ sin (θ) Substitute again: Energy at C: mg d ]^$ sin (θ) Substitute in numerical values: Energy at C: (0.5019)(9.8) d ]^$ sin (13.163) Simplify: Energy at C: d ]^$

16 Appendix D: Finding k of the linearly stretched rubber band (not pinched) The rubber band was initially stretched in one direction using the Vernier Force Sensor, and the force was recorded in Newtons through Logger Pro. The band was stretched to the 30, 40, 50, 60, and 70 cm markings on the meter stick, but 19.9 cm, the length of the slack rubber band, was subtracted from each of the total stretches to determine the Δx. As a result, the new values became , 30.1, 40.1, and 50.1 cm. The data was then graphed with the Force on the y axis and the Δx on the x axis in Excel, and then the linear line of best was found. The slope of this line, _ Δx, was then found, which is equivalent to the value of k. Force (N) y = 8.144x R² = Finding the value of k Δx (m) F (N) IV IV IV IV IV Graph of Force vs Δx of Straight Rubber Band k = kg / s Δx (m)

17 Appendix E: Graph of Force vs Δx of Pinched Rubber Band y = x x R² = Force (N) Δx (m)