The ballistic pendulum

Transcription

1 (ta initials) first nae (print) last nae (print) brock id (ab17cd) (lab date) Experient 3 The ballistic pendulu Prelab preparation Print a copy of this experient to bring to your scheduled lab session. The data, observations and notes entered on these pages will be needed when you write your lab report. Copile these printouts to create a lab book for the course. Perfor the following tasks as indicated. Then answer the following questions and subit the in your Ballistic Pendulu Prelab assignent, due at Turnitin the day before you perfor your experient in the lab. Turnitin will not accept subissions after the due date. Unsubitted prelab reports are assigned a grade of zero. The Ballistic Pendulu Prelab assignent teplate is found at the Lab Docuents page at the course website: DAgostino/lab-anual 1. Video Introduction Watch the video introduction. 2. To get a feel for the concepts behind the calculations you will have to do in analyzing this experient, visit the following site and try a few saple calculations for yourself. You ay wish to first work through the introductory theory portion of the lab write-up below. (Don t worry about sharing the results with your instructor; the purpose of this is to failiarize yourself with the concepts.) You ay also wish to play with the following siulation, which ay help you get a feel for how changing the values of the colliding asses and other initial paraeters changes the final values of soe of the final paraeters Read the relevant sections of the textbook to failiarize yourself with the following concepts. (a) oentu (b) kinetic energy (c) gravitational potential energy (d) elastic potential energy (e) echanical energy (f) elastic and inelastic collisions 17

2 18 EXPERIMENT 3. THE BALLISTIC PENDULUM (g) the principle of conservation of oentu (h) the principle of conservation of energy 4. Suarize in a few sentences how to deterine the initial speed of the projectile and the stiffness constant of the spring in the launcher using the easureents that you will ake in this experient. What is expected here is a conceptual explanation of your approach, not specific forulas. 5. Read through the rest of the lab instructions for this experient in this docuent.! Iportant! Be sure to have every page of this printout signed by a TA before you leave at the end of the lab session. All your work needs to be kept for review by the instructor, if so requested. CONGRATULATIONS! YOU ARE NOW READY TO PROCEED WITH THE EXPERIMENT! Theory In this experient we explore the transfer and conservation of energy and oentu in a collision of two objects. One of these objects is a sall projectile of ass that is projected at a certain speed v by a launcher. The second object is a pendulu that is initially stationary. As the projectile hits the pendulu, a re-distribution of energy and oentu takes place. In a certain class of collisions, the projectile is captured by the pendulu. For such inelastic collisions, we can consider the pendulu and projectile to be one single object after the collision, and this one cobined object carries all of the kinetic energy K after and oentu P after after the collision. If the ass of the pendulu is M then the total ass of the cobined object after the collision is = M +. If the pendulu is stationary when the projectile hits, the pendulu contributes Figure 3.1: Ballistic Pendulu nothing to the total kinetic energy and oentu of the syste before the collision. Thus the principle of conservation of oentu in this case yields (where v T is the speed of the cobined object iediately after the collision): Solving Equation 3.2 for v T, we obtain P before = P after v = v T (3.1) v = (M + )v T v T = v (3.2) Using the expression for the speed of the cobined object iediately after the collision fro the previous equation, we can show that the kinetic energy of the cobined object iediately after the collision is less than the kinetic energy of the projectile just before the collision: K after = 1 2 v 2 T

3 19 K after = 1 ( ) v 2 2 K after = 1 ( 2 2 M v 2 ) T MT 2 K after = 1 ( 2 v 2 ) 2 K after = ( ) 1 2 v2 K after = K before Because the ass of the projectile is less than the ass of the cobined object after the collision, it follows that and therefore < 1 K after < K before As the pendulu begins to swing after the collision, the kinetic energy of the cobined object is gradually converted to gravitational potential energy as the pendulu rises. At the peak of its swing, the pendulu s echanical energy is all in the for of gravitational potential energy, and its kinetic energy is zero. As the pendulu begins to ove downwards after it reaches the peak of its otion, its gravitational potential energy is gradually converted to kinetic energy as the pendulu falls. At the lowest point of its otion, the pendulus s echanical energy is all in the for of kinetic energy, and its graivational potential energy can be considered to be zero. Thus, we can write E top = gh (3.3) and E botto = 1 2 v 2 T (3.4) Assuing that echanical energy is conserved during the initial swing of the pendulu, we can equate the expressions in Equations 3.3 and 3.4. While doing this, we can cobine Equations 3.2 and 3.4 to eliinate v T to obtain an expression that relates the initial speed v of the projectile to the final elevation h of the cobined object, as follows. 1 2 vt 2 = gh ( ) 1 v 2 2 = gh ( M v 2 ) T MT 2 = gh ( 1 2 v 2 ) = gh 2 ( 2 v 2 ) = 2 gh

4 20 EXPERIMENT 3. THE BALLISTIC PENDULUM 2 v 2 = 2MTgh 2 ( M v 2 2 ) = T 2gh v = 2 2gh Equation 3.5 can be expressed in ters of the axiu angle of the pendulu s otion, as follows. The length Rc describes the radius of the arc fro the pivot point to the centre of ass of cobined rod, block, and block contents. With the vertical orientation of R c as the base of a right-angled triangle, h can be expressed in ters of the axiu angle θ of the swing: Rc cos θ = (Rc h). Solving for h, we obtain h = Rc Rc cos θ = Rc (1 cos θ) (3.5) Inserting this expression for h into Equation 3.5, we obtain v = 2gRc (1 cos θ) (3.6) There is another energy conversion taking place, even before the collision. In the experient, the launcher transfers soe of its elastic potential energy into the projectile s initial kinetic energy. (Siilarly, the force exerted by the spring on the projectile provides the projectile s initial oentu.) Applying the principle of conservation of energy to the transfer of energy fro the spring to the projectile yields (where x is the axiu displaceent of the spring fro its equilibriu position and k is the stiffness constant of the spring) elastic potential energy of spring = initial kinetic energy of projectile 1 2 kx2 = 1 2 v2 k = v2 x 2 (3.7) Calculating v using Equation 3.5 and easuring the value of x allows us to use Equation 3.7 to deterine the stiffness constant of the spring. Note that there is a an additional coplication: The spring has a pre-load displaceent. That is, when the spring is in its relaxed state, there is soe copression in the spring. Think about how the previous analysis has to be odified to account for this pre-load copression. Procedure The launcher coponent of the ballistic pendulu consists of a sliding etal rod surrounded by a precision spring. When the lever that is connected to the sliding rod is pulled to the left, the spring is copressed and the rod extends out of the left side of the launcher. This extension can be easured to deterine the distance that the spring was copressed. There are four slots on the side of the launcher. The lever is oved fro its relaxed position at the rightost slot of the launcher and is then lowered into one of the other three slots to lock the spring at one of three copression settings that we will nae, fro right to left: short, ediu, or long range. A cylindrical barrel on the right side of the launcher holds the projectile, a steel ball. As the lever is slowly raised in the slot, the launcher suddenly discharges with the spring extending and pushing the rod to strike the ball. The ball exits the launcher and ipacts the pendulu. The pendulu consists of a rod and bob of cobined ass M attached to a pivot point. When an ipact takes place, the pendulu catches the ipacting ass, changing the total ass of the pendulu

5 21 to = M +, and is caused to swing about the pivot point to a axiu angle of deviation θ, relative to the initial vertical position of θ = 0. The pendulu drags with it a pointer that stops at the liit of the swing and identifies the value of θ on a degree scale concentric with the pivot. The pendulu then free falls back to the vertical position to stop against the barrel. A sall aount of friction between the pointer and the scale prevents the pointer fro falling back along with the pendulu. The effect of this friction and the ass of the pointer on the syste is negligible.? The pointer, initially at rest, is accelerated along with the the pendulu ar on ipact. Could it keep oving past the liit of the pendulu ar, after the ar has stopped, and thus give inaccurate angle readings? Note: when the launcher is in the discharged position, the spring is not fully extended but is subjected to a copression preload x 0 and is thus storing soe potential energy. You will deterine this preload at the end of the experient. Caution: Do not place ball in the launcher until the lever is properly lowered and secure in a slot, otherwise an unintended spring discharge could occur, ejecting the ball and possibly cause an injury. Data gathering and analysis To deterine the physical characteristics of the ballistic pendulu apparatus: 1. reove the pendulu ar fro the ballistic pendulu assebly by unscrewing the pivot screw. Replace the screw for safekeeping; 2. easure with a digital scale (σ = ±0.01g) the ass of the ball and ball/pendulu assebly ; =... ±... kg =... ±... kg 3. deterine the centre of ass point of the pendulu/ball cobination by balancing it on the edge of the steel ruler and noting the position of the balance point on the scale located on the pendulu ar, then easure with the ruler the distance fro this point to the centre of the pendulu pivot (see scheatic Figure 4.2). 4. The centre-of-ass distance R c, fro the balance point to the centre of the pivot hole on the ar of the pendulu of ass, is R c =... ± With the launcher discharged, easure the distance in that the rod extends fro the left end of the launcher. You will subtract this rod offset fro all of the following easureents. The offset is offset =... ± Copress and lock the spring at the short range setting. Measure the distance that the rod now extends fro the end of the launcher. Subtract fro this length the rod offset and record the result below and as x in Table 3.2: x short =... ±....

6 22 EXPERIMENT 3. THE BALLISTIC PENDULUM Figure 3.2: Generic diagra for deterining a pendulu centre-of-ass 7. Repeat the above step for the ediu and long range settings: x ediu =... ±... x long =... ± Deterine the easureent error in the angle θ of the protractor scale used by the launcher θ = ± and finally, replace the pendulu ar, aking sure that the pivot screw is tight. Before you begin to gather ballistic data reeber to avoid sitting directly in front of the discharge path of the launcher barrel and CAUTION: Always wear safety glasses while using the launcher. 1. Move the lever fro the discharged position to the first slot and push it down to lock the spring at the short range setting. Further copression selects the ediu range and finally, the long range setting. 2. Load the launcher by raising the pendulu and placing the ball at the end of the barrel, then lower the pendulu to freely hang in the vertical position. 3. Hold the pendulu in the vertical position and ove the angle indicator to the 0 ark. If the indicator does not reach zero, you will need to subtract this difference fro all your angle readings. 4. To fire the launcher, gently raise the lever fro the retaining slot to release the spring.! Note: if the ball falls out of the pendulu when launched, the assebly is not level or the pendulu bob position needs adjustent. Make sall adjustents to the base leveling screws to centre the pendulu with the launcher barrel or See a TA for assistance.! Note: be sure that the angle indicator does not ove when the pendulu falls back i and strikes the launcher barrel, otherwise you will get an incorrect angle reading. If this happens, gently stop the falling pendulu with your finger before it strikes the launcher. 5. Perfor five short range launches, recording the angle θ i reached in trial i = in the appropriate spaces of Table 3.1. These values should be within ± 0.5 of one another, otherwise redo the set of easureents. 6. Perfor five trials using the ediu and then the long range settings.

7 ! Have a TA check and approve your angle data before proceeding with the calculations. 7. Calculate an average value θ for the five angles θ i obtained in each of the three sets of trials and enter these in Tables 3.1 and 3.2. To avoid soe lengthy standard deviation calculations, let the error in the angle θ i be the easureent error of the angle scale, θ = ± 0.5.? Looking at your data, is this short cut a valid way of estiating the error in θ? 23 range θ 1 θ 2 θ 3 θ 4 θ 5 θ θ short ediu long Table 3.1: Experiental angle values at three force settings You now need to calculate v and v for the three range settings. Note that in Equation 3.6 only the (1 cos θ) ter changes with θ. Chances of error will be iniized if the constant quantities are equated to a ter C and C and are evaluated only once. All the angle error values ust be expressed in radians. C = 2gRc ( MT ) C 2 C = + ( ) 2 + ( ) 2 Rc 2R c v s = C ( C ) 2 ( sin θ θ (1 cos θ) v s = v s + C 2cos θ ) 2 v s =... ±... /s? What should be the diensions of C? And those of C/C? Calculate the axiu kinetic energy of the steel ball at the oent that it lost contact with the launcher rod and enter the value in Table 3.2. Show a coplete calculation for the short range setting: K s = 1 2 v2 s K s = K s ( ) 2 ( + 2 v ) 2 s v s

8 24 EXPERIMENT 3. THE BALLISTIC PENDULUM K s =... ±... J If no energy is lost (or gained) during the interaction, the kinetic energy K is equal to the potential energy range θ v (/s) x () x 2 ( 2 ) K (J) short ± ± ± ± ± ediu ± ± ± ± ± long ± ± ± ± ± Table 3.2: Paraeters for the calculation of the kinetic energy K and the force constant k V = kx 2 /2 released by the spring when the ball was discharged, K = kx 2 /2. This is the equation of a straight line Y = MX with Y = K, X = x 2 and slope M = k/2. Plotting K as a function of x 2 yields fro the slope a value for the spring constant k of the launcher spring. Shift focus to the Physicalab software and enter in the data window the three data pairs and corresponding errors as four space-deliited nubers: x 2 K K x 2. Select scatter plot. Click Draw to generate a graph of your data. Your graphed points should well approxiate a straight line.! Unless the three points are reasonably collinear, the fit will not yield a valid result. You need to deterine and correct the source of the error before proceeding. Consult the TA if you are stuck. Select fit to: y= and enter A*x+B in the fitting equation box. Click Draw to perfor a linear fit of the data. Label the axes and include a descriptive title. Click Send to: to eail yourself a copy of the graph for later inclusion in your lab report. Suarize the values for the slope, the Y-intercept Y (X=0) and their associated errors, then calculate a value for k and k: slope =... ±... Y (X=0) =... ±... k =... =... =... k =... =... =... k =... ±...

9 25? The spring constant k is typically expressed in units of Newtons/etre. Using diensional analysis, verify that your diensions for k obtained fro the graph agree with those of N/. Fro the slope and Y-intercept, calculate the corresponding X-intercept at Y=0: X =... =... =... X =... =... =... X =... ± Fro this result estiate the spring preload distance x 0 : x 0 =... =... =... x 0 =... =... =... x 0 =... ±... IMPORTANT: BEFORE LEAVING THE LAB, HAVE A T.A. INITIAL YOUR WORKBOOK! Lab report Go to our web page to access the online lab report teplate for this experient. Coplete the teplate as instructed and subit it to Turnitin before the lab report subission deadline, late in the evening six days following your scheduled lab session. Turnitin will not accept subissions after the due date. Unsubitted lab reports are assigned a grade of zero. Notes:...