Lab 8. Ballistic Pendulum Goals To determine launch speed of a steel ball for short, medium, and long range settings on projectile launcher apparatus using equations for projectile motion. To use concepts of gravitational potential energy and conservation of mechanical energy to determine speed of ball plus pendulum as it first begins to swing away from vertical position after collision. To explore relationships between momentum and kinetic energy of ball as launched and momentum and kinetic energy of ball plus pendulum immediately after ball is caught by pendulum apparatus. Introduction The ballistic pendulum carries this name because it provides a simple method of determining speed of a bullet shot from a gun. To determine speed of bullet, a relatively large block of wood is suspended as a pendulum. The bullet is shot into wooden block so that it does not penetrate clear through it. This is a type of sticky collision, where two masses (bullet and block) stick to one anor and move toger after collision. By noting angle to which block and bullet swing after collision, initial speed can be determined by using conservation of momentum. This observation incorporates some predictions that we can check. In this experiment, ballistic pendulum apparatus will be used to compare momentum of steel ball before collision to momentum of ball and pendulum apparatus, equivalent to wooden block plus bullet, after collision. A comparison of kinetic energy of ball before collision with kinetic energy of system afterward will also be made. Figure 8.1 shows a diagram of ballistic pendulum apparatus. For ballistic pendulum experiment, projectile launcher from projectile motion laboratory is mounted horizontally so that pendulum can catch emerging steel ball. The angle indicator can be used to measure maximum angle reached by pendulum as it swings after collision. The angle indicator should read close to zero when pendulum is hanging in vertical position. If reading is measurably different from zero, n take difference in angle readings (maximum angle reading minus initial angle reading). Warning: Never look down launcher barrel. Wear eye protection until everyone is finished launching projectiles. 71
momentum values based on uncertainties of measured horizontal distance traveled and CHAPTER measured 8. vertical BALLISTIC height. PENDULUM The momentum of ball-pendulum system before ball 72 collides with pendulum is now known. angle indicator plumb bob launcher Figure 8.1. Ballistic pendulum apparatus. Momentum of steel ball before collision 3. Momentum of ball and pendulum after collision The speed (and from it momentum) of ball and pendulum just after collision is For this part of experiment, remove pendulum by gently unscrewing rod that supports computed by assuming that kinetic energy of ball and pendulum just after collision is totally its upper converted end. Now into determine gravitational muzzle potential velocity energy of at steel top ball of its by swing. firing it This horizontally requires that and measuring forces distance on traveled ball and horizontally pendulum before system striking during ground. swing are Dosmall this for (negligible). short, medium, The frictional increase and longin range gravitational settings potential of launcher. energy is The just momentum weight of pendulum ball is found times by multiplying change in its height, mass times and its velocity. change in Quantitatively height can be estimate computed from uncertainties maximum in se angle momentum of pendulum values based swing on and uncertainties some straightforward of measured trigonometry. horizontalsince distance traveled pendulum andis not measured a point mass, vertical height. change The momentum in potential ofenergy ball-pendulum is given by system change before in height ball of its collides center with of gravity. pendulum The center is now of gravity known. can be located by removing it from its support screw at top and n balancing it on a thin knife edge. (A thin ruler works.) While you have pendulum disassembled, be sure to measure mass of pendulum and distance from pivot point at top to center of Momentum gravity. of ball and pendulum after collision Mount The speed (and pendulum from it so that momentum) it will catch ofand ball trap and steel pendulum ball before just after proceeding. collision Be is gentle computed as you by assuming screw that pendulum kinetic support energy rod; ofit does ball not and need pendulum to be tight. just Now afterlaunch collision ball is into totally pendulum converted from into gravitational short, medium, potential and energy long at range top settings of itsof swing. projectile This requires launcher. that Repeat frictional each forces measurement on ball and several pendulum times and system take during appropriate swing averages. are small (Remember (negligible). to check The increase initial in angle of pendulum at rest.) gravitational potential energy is just weight of pendulum times change in height, and change in height can be computed from maximum angle of pendulum swing and some straightforward trigonometry. Since pendulum is not a point mass, change in potential energy is given by change in height of its center of gravity. The center of gravity can be located by removing it from its support screw at top and n balancing it on a knife edge. (A thin ruler works.) While you have pendulum disassembled, be sure to measure mass of pendulum and distance from pivot point56 at top to center of gravity.
CHAPTER 8. BALLISTIC PENDULUM 73 Mount pendulum so that it will catch and trap steel ball before proceeding. Be gentle as you screw in pendulum support rod; it does not need to be tight. Now launch ball into pendulum using short, medium, and long range settings of projectile launcher. Repeat each measurement several times and take appropriate averages. (Remember to check initial angle of pendulum at rest.) From your data calculate speed of pendulum and ball toger just after collision. Multiply by appropriate mass to get momentum. The momentum of ball-pendulum system just after collision is now known. Is momentum conserved? Compare initial momentum of ball and pendulum system before collision with final momentum of same system just after collision using your calculated velocities and measured masses just before and just after collision. Is momentum conserved? You cannot answer this question without comparing difference between two momenta with uncertainty of this same difference. If difference between momenta is more than three times uncertainty of difference, odds of difference being due to random variations is small your data do not support conservation of momentum in this case. If you expect momentum to be conserved, examine your calculations and procedures for errors. Is kinetic energy conserved? Since you know masses and speeds of objects before and after collision, you can calculate kinetic energies of system before and after collision. Is kinetic energy conserved? To answer this question, you will need to estimate your experimental uncertainties and compare m with any observed differences, as you did to test conservation of momentum. Assuming that momentum is conserved before and after collision, find a general symbolic mamatical expression for ratio of final kinetic energy over initial kinetic energy. You may need some help from your TA here. Using data from your earlier calculations, compare your experimental kinetic energy ratio to that predicted by assuming momentum is conserved. Is it same ratio? Is overall energy conserved in this collision? If so, what forms of energy would need to be included to satisfy general energy conservation principle? Note: A simplification has been made by assuming that pendulum consists of a point mass on end of a string whose length is equal to distance from pivot point to its center of mass. When pendulum swings, it necessarily rotates about its center of mass. This suggests that some rotational kinetic energy is imparted to ball and pendulum system along with its translational kinetic energy (mv 2 /2). If significant, this would produce a systematic error in calculated speed of ball and pendulum system after collision. Would it make calculated speed too high or too low? Can you detect any systematic error in your calculated values? Discuss.
CHAPTER 8. BALLISTIC PENDULUM 74 Summary Mechanical energy and momentum are conserved only when certain conditions are met. Qualitatively summarize your results, explaining why collision between ball and pendulum conserves momentum but not mechanical energy. Similarly, explain why motion of pendulum during its swing conserves mechanical energy but (apparently) not momentum. AA extract information from representation correctly Grading Rubric No Effort Progressing Expectation Exemplary No visible attempt is made to extract information from experimental setup. Information that is extracted contains errors such as labeling quantities incorrectly, mixing up initial and final states, choosing a wrong system, etc. Physical quantities have no subscripts (when those are needed). Most of information is extracted correctly, but not all of information. For example physical quantities are represented with numbers and re are no units. Or directions are missing. Subscripts for physical quantities are eir missing or inconsistent. All necessary information has been extracted correctly, and written in a comprehensible way. Objects, systems, physical quantities, initial and final states, etc. are identified correctly and units are correct. Physical quantities have consistent and informative subscripts. AB construct new representations from previous representations construct a different representation. Representations are attempted, but omits or uses incorrect information (i.e. labels, variables) or representation does not agree with information used. Representations are constructed with all given (or understood) information and contain no major flaws. Representations are constructed with all given (or understood) information and offer deeper insight due to choices made in how to represent information. AC evaluate consistency of different representations and modify m when necessary made to evaluate consistency. At least one representation is made but re are major discrepancies between constructed representation and given experimental setup. There is no attempt to explain consistency. Representations created agree with each or but may have slight discrepancies with given experimental representation. Or re is inadequate explanation of consistency. All representations, both created and given, are in agreement with each or and explanations of consistency are provided. Labs: 1-11 AE Force Diagram constructed. Force Diagram is constructed but contains major errors such as mislabeled or not labeled force vectors, length of vectors, wrong direction, extra incorrect vectors are added, or vectors are missing. Force Diagram contains no errors in vectors but lacks a key feature such as labels of forces with two subscripts vectors are not drawn from single point, or axes are missing. The diagram contains no errors and each force is labeled so that it is clearly understood what each force represents. Vectors are scaled precisely.
CHAPTER 8. BALLISTIC PENDULUM 75 AF Sketch Labs: 2, 3, 6-8, 11, 12 No Effort Progressing Expectation Exemplary constructed. Sketch is drawn but it is incomplete with no physical quantities labeled, or important information is missing, or it contains wrong information, or coordinate axes are missing. Sketch has no incorrect information but has eir a few missing labels of given quantities. Subscripts are missing or inconsistent. Majority of key items are drawn. Sketch contains all key items with correct labeling of all physical quantities have consistent subscripts; axes are drawn and labeled correctly. AG Mamatical Labs: 1-4, 6-12 constructed. Mamatical representation lacks algebraic part ( student plugged numbers right away) has wrong concepts being applied, signs are incorrect, or progression is unclear. No error is found in reasoning, however y may not have fully completed steps to solve problem or one needs effort to comprehend progression. Mamatical representation contains no errors and it is easy to see progression of first step to last step in solving equation. The solver evaluated mamatical representation with comparison to physical reality. CA identify hyposis to be tested No mention is made of a hyposis. identify hyposis to be tested but is described in a confusing manner. The hyposis to be tested is described but re are minor omissions or vague details. The hyposis is clearly stated. CC make a reasonable prediction based on a hyposis No prediction is made. The experiment is not treated as a testing experiment. A prediction is made but it is identical to hyposis OR Prediction is made based on a source unrelated to hyposis being tested OR is completely inconsistent with hyposis being tested OR Prediction is unrelated to context of designed experiment. Prediction follows from hyposis but is flawed because relevant experimental not considered and/or prediction is incomplete or somewhat inconsistent with hyposis and/or prediction is somewhat inconsistent with experiment. A prediction is made that follows from hyposis, is distinct from hyposis, accurately describes expected outcome of designed experiment, incorporates relevant assumptions if needed. CD identify assumptions made in making prediction identify any assumptions. identify assumptions, but irrelevant or are confused with hyposis. Relevant identified but are not properly evaluated for significance in making prediction. Sufficient assumptions are correctly identified, and are noted to indicate significance to prediction that is made.
CHAPTER 8. BALLISTIC PENDULUM 76 CE determine specifically way in which assumptions might affect prediction CF decide wher prediction and outcome agree/disagree CG make a reasonable judgment about hyposis GD record and represent data in a meaningful way GE analyze data appropriately IA conduct a unit analysis to test self-consistency of an equation No Effort Progressing Expectation Exemplary determine effects of assumptions. No mention of wher prediction and outcome agree/disagree. No judgment is made about hyposis. Data are eir absent or incomprehensible. analyze data. No meaningful attempt is made to identify units of each quantity in an equation. The effects of mentioned but are described vaguely. A decision about agreement/disagreement is made but is not consistent with outcome of experiment. A judgment is made but is not consistent with outcome of experiment. Some important data are absent or incomprehensible. They are not organized in tables or tables are not labeled properly. analyze data, but it is eir seriously flawed or inappropriate. identify units of each quantity, but student does not compare units of each term to test for self-consistency of equation. The effects of determined, but no attempt is made to validate m. A reasonable decision about agreement/disagreement is made but experimental uncertainty is not properly taken into account. A judgment is made, is consistent with outcome of experiment, but not taken into account. All important data are present, but recorded in a way that requires some effort to comprehend. The tables are labeled but labels are confusing. The analysis is appropriate but it contains minor errors or omissions. check units of each term in equation, but student eir mis-remembered a quantity s unit, and/or made an algebraic error in analysis. The effects of determined and validated. A reasonable decision about agreement/disagreement is made and experimental uncertainty is taken into account. A judgment is made, consistent with experimental outcome, and taken into account. All important data are present, organized, and recorded clearly. The tables are labeled and placed in a logical order. The analysis is appropriate, complete, and correct. The student correctly conducts a unit analysis to test selfconsistency of equation.
CHAPTER 8. BALLISTIC PENDULUM 77 EXIT TICKET: Quit Capstone and any or software you have been using. Straighten up your lab station. Put all equipment where it was at start of lab. Report any problems or suggest improvements to your TA. Have TA validate Exit Ticket Complete.