Instruction Sheet Martin Henschke, Ballistic Pendulum art. no.:

Transcription

1 Physics Educational Tools Dr. Martin Henschke Gerätebau Dieselstr. 8, 5374 Erftstadt, Gerany Instruction Sheet Martin Henschke, Ballistic Pendulu art. no.: 6573 Copyright 6 Martin Henschke Gerätebau Technical alterations reserved Fig. 1: Coponents 1 Ball launcher (6567) Back plate 3 Guide for swing pointer 4 Bearing screw 5 Counter bearing 6 Swing pointer 7 Angle scale 8 Pendulu 9 Ball catcher 1 Base plate 11 Table clap 1 nurled screw 13 Rarod (delivered with 6567) 14 Extra weights, pcs. 1

2 1. Safety instructions This instruction sheet is concerned ainly with the ballistic pendulu. You should also read the instructions for the ball launcher To check whether a ball is located in the ball launcher and the spring is cocked, only use the observation holes at the sides. Do not look into the barrel fro the front. Risk of injury! Never ai at people! Protective goggles should be worn during the experients. The ball launcher should always be stored with the spring loose and with no ball in the barrel.. Description The ballistic pendulu is for experient-based deterination of the launch velocity of a ball when it leaves the ball launcher. It is also possible to deterine trajectories when the ball is launched horizontally or at an angle. Launch heights of 5, 1, 15, or 3 c can be selected easily with the aid of the drilled holes. Thanks to the extree lightness of the pendulu, the experient can be perfored using coparatively safe plastic balls instead of steel balls. Experients involving inelastic collisions (quantitatively) and elastic collisions (qualitatively) can be evaluated. The velocity of the balls deterined fro trajectory and pendulu experients typically agree to within about 3%. Extra weights allow various pendulu travels to be investigated for constant speeds. 3. Operation and aintenance First the ballistic pendulu is screwed to a stable bench by eans of its clap. The ball launcher is then screwed to the back plate () fro behind either in a horizontal position in front of the pendulu as in Fig. 1 or as shown in Fig 3. Tip: if the workbench is not stable enough, it ay be that when the pendulu swings to its axiu extent and then swings back, it ay jog the apparatus when striking the ball launcher, causing the swing pointer to be shifted out of line. If this happens, the pendulu should rather be stopped by hand. Balls should always be loaded when the spring is not under tension by placing the sphere in loosely through the front of the plastic cylinder within the device. The sphere is then pushed down inside the barrel using the rarod until the desired spring tension has been reached. The rarod should not be reoved too quickly, otherwise the suction its reoval produces ay pull the sphere out with it. The position of the sphere ay only be checked using the observation holes. Never look into the barrel! Before launching, ensure that no one is in the way of the trajectory. To launch, the cord of the launching lever is briefly pulled perpendicularly to the lever. The pendulu (8) can be reoved by undoing the bearing screw (4) and turned by 18 so that it is installed with the rear of the ball catcher (9) pointing towards the launcher (experients on elastic collision). The counter bearing (5) is designed so that the pendulu hangs at a slight angle if the bearing screw is only light tightened. This eans that the ball

3 catcher is not precisely in front of the launch aperture of the launcher. For this reason, the bearing screw should be tightened until the catcher and the launch aperture are in line. After turning the pendulu round, or if necessary, the guide (3) for the swing pointer (6) should be adjusted so that the pointer just touches it when the pendulu is suspended at rest. The screw on the guide should only be finger-tightened to avoid the appearance of pressure on the pendulu rod. Maintenance: the ballistic pendulu principally requires no aintenance. If necessary soe non-acid grease (Vaseline) can be applied to the bearing screw (4) and the knurled screw (1). Other than in the vicinity of the scale, the apparatus ay be cleaned using acetone, ethanol (white spirit) or petroleu ether as required. Avoid suberging the equipent in water. 4. Experient procedure and evaluation 4.1 Ballistic pendulu Experient setup The experient setup corresponds to Fig. 1 for experients on inelastic collision. For experients on elastic collisions, the pendulu should be turned round by 18 (cf. Section 3 Operation ) Experient procedure It is practical for these experients to enter the experient nuber, the spring tension (1, or 3), the type of collision (inelastic i or elastic e ), the nuber of extra weights used and the easured angle ϕ. In order to obtain the ost accurate experient results, after one shot, a second should be perfored with the swing pointer not having been reset to in between. This iniizes the unavoidable frictional losses of the swing pointer. Exaple experient sequence: No Spring Type of Extra weights Angle ϕ tension collision 1 1 i 17.5 i i i i i e e e 6. 3

4 4.1.3 Experient evaluation Inelastic collision The following equation is valid for the swinging pendulu due to conservation of energy E = (1) pot E kin where the potential energy is E = g h () pot Here is the al ass of the pendulu including the ball and any extra weights, g is the acceleration due to gravity and h is the difference in height of the center of gravity of the pendulu at rest and at the axiu extent of its swing. Fro the easured angle ϕ and the easured length l S to the center of gravity according to Fig. the following is derived: h = ( 1 cosϕ) (3) l S Fig. : Deterining the required lengths. Distance between center of gravity and axis of rotation ( l S ) should be easured including the ball and any additional weights when the collision is inelastic. To perfor the easureent, the pendulu ay, for exaple, be balanced on a ruler ounted on its side. The distance between the center of the ball and the axis of rotation is l = 8. The kinetic energy can be calculated fro the oent of inertia Ι relative to the axis of rotation and the axiu angular speed ω according to the equation 1 Ekin = Ι ω. (4) If Equations and 4 are inserted into Equation 1 and h eliinated using Equation 3 then the equation can be rearranged to: g l S (1 cosϕ) ω = (5) Ι However, we are not seeking ω, but the initial velocity of the ball v. The relationship between the two values is given by the equation for the conservation of angular oentu directly before and after the collision: L = L (6) with the angular oentu of the ball L = lv (7) before the collision and the al angular oentu L = Ιω (8) after the collision. Inserting Eqs. 7 and 8 into Eq. 6 gives: lv = Ιω (9) 4

5 Resolving this for ω and equating with Eq. 5 leads to the following relationship v 1 = Ι g ls(1 cosϕ) l (1) The oent of inertia is in principle deterined fro the integral Ι = l d (11) where l is the distance of each ass eleent d fro the axis of rotation. Since in this case it is not the oent of inertia that we seek to derive Ι can also be calculated fro the period T of the pendulu (with ball and any extra weights). For a physical pendulu the following is valid for sall deflections 1 : T Ι = g ls (1) π This eans that all the variables are now known or calculable. For the above exaple, the following table eerges: No / kg / kg l S / T / s v in /s The nueric values should be deterined separately for every pendulu, since aterial and anufacturing tolerances ean that values ay differ fro one to another Elastic collision For a swinging pendulu Eq. 5 is still valid for the otion after a collision, the only difference being that the oent of inertia Ι P is deterined without the ball but with any extra weights (pendulu ass P ): g (1 cosϕ) Ι P l ω = S (13) P To deterine the relationship between ω and the initial velocity v both the conservation of angular oentu and the conservation of energy before and after the collision ust now be used. The additional equation is required since the syste has an additional degree of freedo in the ball velocity v after the collision. As for Eq. 9, the following is true for the angular oentu: l v = l v + Ι ω v = v ΙPω l P If this velocity v is inserted into the equation for the conservation of energy v = v + ΙPω (15) (14) 1 Recknagel, A.: Physik Mechanik, 3te Auflage, VEB Verlag Technik Berlin,

6 by rearranging in various steps the following expression is obtained for v 1 ΙP v = ω l 1+ (16) l If Eq. 13 is plugged in here and Ι P deterined as in Eq. 1, then v can be calculated for an ideal inelastic collision: No / kg P / kg l / T / s v S in /s These values for v are about 18% saller than those obtained for inelastic collisions. This can be explained by the fact that the elastic collisions are not entirely ideal. 4. Deterination of trajectories Experient setup One possible experient setup is shown scheatically in Fig. 3 (not to scale). The drill holes in the back plate of the pendulu are placed so that when a ball is fired to land directly on the workbench, the launch heights are 5, 1, 15, and Fig. 3: Experient setup, key: 1 Ball launcher, Launch position of the ball, 3 Paper, 4 Carbon paper, 5 Easel with whiteboard (for exaple) 6

7 When launching against a vertical wall (e.g. whiteboard) the radius of the ball (1.5 c) should be subtracted fro the distance between the point of launch and the wall to obtain the distance easureent x M. The height easureent y M relative to the launch height is given by the height of the ipact on the wall inus 6.5, 11.5, 16.5, 1.5 or 31.5 depending on the hole used Experient procedure It is practical for these experients to note the experient nuber, the spring tension (1, or 3), the launch angle and the values x M and y M. Exaple: No Spring tension launch angle ϕ / distance x M / c target height y M / c Experient evaluation It is practical to take as the origin of the coordinate syste the id-point of the ball at the oent of launch. Then the following applies: v x = cosϕ (17) v v y = v sinϕ (18) 1 y = v t g t y (19) x = v t () x Fro Eq. = x v x, whereby the tie can be eliinated fro Eq.19. If x t / v and v y are then eliinated fro the resulting equation using Eqs. 17 and 18, the following is obtained g y = x tanϕ x (1) v cos ϕ This is the equation for the trajectory. In this equation only the launch velocity v is unknown since the distances x and y were easured during the course of the experients. If v is calculated for the various experients, the following results are obtained: Spring tension v in /s The nubers are based on a al of 5 experients, of which only 6 are explicitly listed in the above table. The trajectory can now be obtained fro these using Eq. 1 and copared to the easured values. The result is shown in Fig. 4. 7

8 -,1 y / -, -,3 -,4,5 1 1,5 x / Fig. 4: Coparison of easureents and calculated curve, x = horizontal ball distance, y = vertical height, sybols = easured values (circles = spring tension 1, squares = spring tension, rhobuses = spring tension 3), lines = calculated trajectories 8