The ballistic pendulum

Transcription

1 (ta initials) first nae (print) last nae (print) brock id (ab17cd) (lab date) Experient 4 The ballistic pendulu In this Experient you will learn how to deterine the speed of a projectile as well as the energy of the launcher that fired it that real-world interactions typically involve a series of energy exchanges the energy relationship between the linear otion of a ball and the angular otion of a pendulu that successful experiental results rely on careful easureent and application of technique to extend your data analysis capabilities with a coputer-based fitting progra; to apply different ethods of error analysis to experiental results. 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 and as reference aterial during your final exa. Copile these printouts to create a lab book for the course. To perfor this Experient and the Webwork Prelab Test successfully you need to be failiar with the content of this docuent and that of the following FLAP odules ( Begin by trying the fast-track quiz to gauge your understanding of the topic and if necessary review the odule in depth, then try the exit test. Check off the box when a odule is copleted. FLAP PHYS 1-2: Errors and uncertainty FLAP MATH 1-1: Arithetic and algebra FLAP MATH 1-2: Nubers, units and physical quantities Webwork: the Prelab Ballistic Pendulu Test ust be copleted before the lab session! Iportant! Bring a printout of your Webwork test results and your lab schedule for review by the TAs before the lab session begins. You will not be allowed to perfor this Experient unless the required Webwork odule has been copleted and you are scheduled to perfor the lab on that day.! Iportant! Be sure to have every page of this printoutsigned 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! 26

2 27 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 given a certain velocity v by a launcher. The second object is a stationary pendulu. 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, there is only one cobined object that is oving at the end, and carries all of the kinetic energy K and oentu P. If the ass of the pendulu is M then the total ass of the cobined object at the end of the collision is M T = M +. Figure 4.1: Ballistic Pendulu If the bob is stationary when the projectile hits, it contributes nothing to the total kinetic energy and oentu of the syste before the collision. Thus the Law of Conservation of Moentu in this case yields: P before = v = P after = (M +)v T = M T v T (4.1) where v T is now the velocity of the cobined object iediately after the collision. Knowing v and M T, we can use Equation 4.1 to deterine v T and show that the kinetic energy is not conserved: K after = 1 2 M Tv 2 T = 1 2 M T ( ) v 2 = M T M T ( ) v 2 = K before < K before. (4.2) 2 M T As the pendulu begins to swing after the collision, another physical process takes place; a conversion of the kinetic energy of the oving object into into gravitational potential energy as it swings up, losing kinetic energy and gaining potential energy. At the botto of the swing, the pendulu of ass M T has all of its energy in the for of kinetic energy. At the top of the swing, all of the pendulu s energy is converted into gravitational potential energy, and as M T oentarily pauses and reverses its otion, the kinetic energy falls to zero. Thus we can write: E botto = K = 1 2 M Tv 2 T = E top = M T gh. (4.3) Cobining Equations 4.1 and 4.3 to eliinate v T gives us an expression that relates the initial velocity v of the projectile to the final elevation h of the cobined object. v = M T 2gh, (4.4) The length R c 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 angle θ of the swing: Rcosθ = (R h). Equation 4.4 then becoes v = M T 2gRc (1 cosθ) (4.5) There is another energy conversion taking place, even before the collision. In the experient, the launcher gives the projectile its initial kinetic energy and oentu by releasing the potential energy

3 28 EXPERIMENT 4. THE BALLISTIC PENDULUM and oentu of a copressed spring an converting it into the kinetic energy of the oving projectile. Conservation of energy in this case yields E initial = PE = 1 2 kx2 = E final = K = 1 2 v2 (4.6) where x is the spring displaceent fro its equilibriu (relaxed) length and k is the spring constant. As the projectile is released fro rest, the projectile has no initial kinetic energy. As the projectile begins to accelerate, a conversion of energy takes place where the potential energy stored in the spring is converted into the kinetic energy of the projectile. At the point where the projectile releases fro the spring, all of the spring s stored potential energy has been converted into the projectile s kinetic energy. We can then deterine a value for the spring constant k of the launcher by equating the initial potential energy of the spring to the final kinetic energy of the projectile: Procedure k = v2 x 2 (4.7) The launcher coponent of the ballistic pendulu consists of a precision spring encased in an aluinu barrel. One end of the spring is secured to the closed end of the barrel. The other end is attached to a piston that slides along the inside of the barrel. The projectile rests on the face of the piston. A trigger echanis allows the piston to be locked in one of three force settings: short, ediu or long range. When the launcher is in the discharged position, the spring is subjected to a sall copression so that it will not rattle when released. This preload x 0 is in the order of 1. For this experient, we will assue that when the launcher is discharged the potential energy stored in the spring is approxiately zero. 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 to M T = 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 or 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 penduluar, after the ar has stopped, and thus give inaccurate angle readings? 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;

4 29 2. verify that the plastic nut securing the brass weights to the botto of the pendulu ar is tight; 3. easure with a digital scale (σ = ±0.01g) the ass of the ball and ball/pendulu assebly M T ; =...±... kg M T =...±... kg 4. deterine the centre of ass point of the pendulu/ball cobination as shown in Figure 4.2. Make sure that the ar of the pendulu is parallel with the length of the scale.? The pendulu placed on the edge of the easuring apparatus will be unstable and not reain balanced in that position. How then can you precisely deterine the length R c? 5. The centre-of-ass distance R c, the distance fro the edge to the centre of the pivot hole on the ar of the pendulu of ass M T, is R c =...±.... Figure 4.2: Experiental setup for deterining the centre-of-ass of the pendulu 6. With the launcher discharged and the ball reoved, use the scale on the plunger to deterine the offset depth of the face of the piston fro the front end of the barrel. You will need to subtract this offset fro all of the following depth easureents. The offset is offset =...± With the ball reoved, copress the piston until it latches at the short range setting. Measure the depth fro the face of the piston to the front end of the barrel. Subtract fro this length the offset depth of the piston and record the result below and as x in Table 4.2: x short =...± Repeat the above step for the ediu and long range settings: x ediu =...±... x long =...± then deterine the easureent error in the angle θ of the protractor scale used by the launcher θ = ±...

5 30 EXPERIMENT 4. THE BALLISTIC PENDULUM 10. 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. Verify that the C-clap securing the pendulu apparatus to the table is tight.? What do you suppose ight happen if the pendulu apparatus was discharged while not being secured to the table? How would your easureent be affected? What physical law is at work here? 2. Swing the pendulu to 90 and support it in that position with one hand or soeone s assistance. 3. With the other hand, load the ballistic pendulu by placing the projectile in the barrel and pushing it with the plunger rod until the trigger locks. This is the short range setting. Further copression selects the ediu range and finally, the long range setting. 4. Once the launcher is loaded, slowly withdraw the plunger, aking sure that the trigger is indeed locked and that the projectile has not rolled away fro the face of the piston, as this would cause an innacurate firing of the launcher. 5. Gently lower the pendulu to the vertical position, and ove the angle indicator to the 0 ark. If the indicator does not reach zero, you will need to subtract the offset fro all your angle readings. 6. To fire the launcher, gently pull the string upward to release the trigger. 7. Perfor five short range launches, recording the angle θ i reached in trial i = in the appropriate spaces of Table 4.1. These values should be within ± 0.5 of one another, otherwise redo the set of easureents. 8. Perfor five trials using the ediu and then the long range settings.! Have a TA check and approve your angle data before proceeding with the calculations. range θ 1 θ 2 θ 3 θ 4 θ 5 θ θ short ediu long Table 4.1: Experiental angle values at three force settings 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 4.1 and 4.2. To avoid soe lengthy standard deviation calculations, let the error in the angle θ i be the easureent error of the angle scale, θ = ± 0.25.? Looking at your data, is this shortcut a valid way of estiating the error in θ?

6 31 You now need to calculate v and v for the three range settings. Note that in Equation 4.5 only the (1 cosθ) 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 = M T 2gRc ( MT ) C 2 C = + M T ( ) 2 + ( ) 2 Rc 2R c v s = C ( C ) 2 ( sinθ θ (1 cosθ) v s = v s + C 2cosθ ) 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 piston of the launcher and enter the value in Table 4.2. 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 K s =...±... J If no energy is lost (or gained) during the interaction, the kinetic energy K is equal to the potential energy V = kx 2 /2 of the spring before 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.

7 32 EXPERIMENT 4. THE BALLISTIC PENDULUM range θ v (/s) x () x 2 ( 2 ) K (J) short ± ± ± ± ± ediu ± ± ± ± ± long ± ± ± ± ± Table 4.2: Paraeters for the calculation of the kinetic energy K and the force constant k 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 =...±...? The spring constant k is typically expressed in units of Newtons per 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 =...±... 2.

8 33 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 your course hoepage on Sakai (Resources, Lab teplates) to access the online lab report worksheet for this experient. The worksheet has to be copleted as instructed and sent to Turnitin before the lab report subission deadline, at 11:00p 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:...