Ballistic Pendulum. Introduction

Transcription

1 Ballistic Pendulu Introduction The revious two activities in this odule have shown us the iortance of conservation laws. These laws rovide extra tools that allow us to analyze certain asects of hysical systes and to e ale to redict the otion of ojects in the systes without using ore colicated analysis. Even in situations wherein we cannot exactly solve the otion, these laws are incredily useful. For instance, if soeone shows us an incredily colicated device that can seeingly roduce electricity with no energy inut whatsoever, we know not to invest oney in this device, as it ust e a sha since it violates the conservation of energy rincile. However, oth of these conservation laws are theoretical constructs that rarely, if ever, hold 100% true in the real world. The exerients that were run showed roof of this, as the exeriental results did not atch the theoretical odel. Energy is lost as it is transferred fro kinetic to otential, and vice-versa. Moentu was not conserved during the collisions, as they turned out to e neither erfectly elastic or erfectly inelastic. It turns out that there is another law at work that liits these other conservation laws: the Second Law of Therodynaics. The Second Law of Therodynaics and Fig 1: Davinci s drawing showing the iossiility of eretual otion The First Law of Therodynaics tells us that the energy involved in any transfer ust e conserved. This would see to ean that we should never run out of energy and should ay no heed to anyody talking aout an energy crisis. The role is that this is not the only law that governs energy transfers. While the total aount of energy does not change, the second law of therodynaics uts liits on the aount of usale energy that can e transferred. One of the consequences of this law is that the total aount of usale energy that coes out of any rocess will e less than the total aount of energy that went into the rocess. The difference etween the total aount of energy inut and the usale energy outut is exended as waste heat. Take, for exale, a all that is droed fro soe height aove the ground. As it falls, air acts uon the all to slow it down. In doing so, soe of the initial otential energy of the all is converted to greater kinetic energy of the olecules of air, which akes the slightly warer. This rings us to the issue of efficiency, which is a easure of the aount of usale energy that is generated during any tye of transfer. If a transfer is very efficient, then the aount of usale energy that is generated is alost equal to the total aount of energy that went into the transfer. This eans that very little waste energy will e roduced. An inefficient transfer, conversely, is one in which ost of the energy going into the rocess is converted to waste heat. For exale, a fluorescent light ul converts aout 0% of the electrical energy that runs through it into visile light energy. While this ay not sound like a very efficient transfer, it is uch etter than the 5% efficiency of an incandescent light ul, which ost eole use.

2 When discussing the efficiency of a rocess, we have to ake sure and not forget all of the transfers that ight need to take lace in order to get to the one under investigation. A great exale of this occurs when coaring the efficiencies of electric and internal coustion engine owered cars. The efficiency of the electric otor in a car is aout 90%, while the efficiency of the internal coustion engine is only aout 5%. However, these efficiencies are not the only things that need to e considered when coaring the two devices. How is the electricity that charges the car created? Where does the gasoline coe fro that owers the internal coustion engine? What tyes of transission systes does each car have? There are any stes and energy transfers that take lace in getting each tye of car to ove, and each one of these has its own individual efficiency. For instance, the average coal urning electric lant is only aout 30-35% efficient in generating electricity (soe newer natural gas lants are closer to 50-60%). This fact greatly reduces the overall efficiency of an electric car. When we consider the total efficiency, fro getting the energy fro its natural source to the car oving down the highway, we find that the electric car is only aout 0% efficient, while the internal coustion engine autooile is aout half that at 10% 1. Ballistic Pendulu In la this week, we are going to look at a series of energy conversions to see how efficiency works. Figure shows a icture of the allistic endulu that we will use in this activity. The device is quite sile to oerate. Pushing ack the sring-loaded iston on the rojectile section stores otential energy that can e used to roel a all. Once the all has een roelled out of the launcher y ulling the trigger, it collides inelastically with the endulu, thus transferring oentu to it. The endulu then swings uward until all of its kinetic energy is Fig. : Ballistic endulu converted to otential energy. The angle easuring syste on the side of the device stos it at this height, allowing for easureents of the aount of otential energy stored. This otential energy can then e coared to the initial kinetic energy to see how efficient the energy is converted fro one for to another. To find out what the theoretical efficiency of this rocess is, let us take the velocity of the all leaving the gun iston to e v 1 the ass of the all to e, the ass of the endulu to e, and the height to which the all-endulu rises to e h f. With this, the easured efficiency of the energy transfers will e ex PE ( )gh final f (1) KE initial v i To calculate what the theoretical efficiency is, we need to use the conservation of linear oentu. We know that v i is related to velocity of the all and endulu after the collision (v ) y v () i v If the endulu does not lose energy while it is rising, then the kinetic energy at the otto should equal the otential energy at the to, giving us the relationshi 1 gh f v (3)

3 If we sustitute oth Equation and 3 into Equation 1, the theoretical efficiency ecoes theory ( )v v i This equation shows that as the ass of the endulu goes to 0, then the efficiency of this energy transfer should go to 100%. This akes sense, as a assless endulu would have no oentu to change during the collision. Furtherore, as the ass of the endulu ecoes large, the efficiency of the energy transfer would tend toward 0. Again, this akes sense, as an infinitely large endulu would not ove after the collision, thus asoring all of the energy. Reference 1 Energy: Its Use and the Environent, nd Edition y Roger A. Hinrichs, Saunders College Pulishing, Orlando, Activity The activity this week relies on a coercially-availale iece of equient: a allistic endulu. There are several different anufacturers of these devices (Cenco and Pasco are two of the ore oular odels). All of the oerate in asically the sae anner, i.e. a sring loaded gun that shoots a all into a cu attached to the end of a endulu. In order to test our odel, we will first need to know the velocity of the all leaving the end of the lunger (v i ). One way to do this would e to shoot the all off of the end of a level tale and to easure how far the all travels efore it hits the ground. The velocity of the all v i is related to the distance D that the all travels efore it hits the ground y the equation D g v i, H where g is the acceleration due to gravity and H is the initial height of the all aove the ground. The velocity can also e easured using hotogates in the following anner. 1. Place the allistic endulu on the taleto such that the shot all will have an unostructed ath to the floor. Reove the endulu ortion of the equient.. Place the hotogate near the end of the lunger such that the released all will sail through the hotogate oening, with the iddle of the all assing etween the LED light/detector. 3. Place the all on the lunger and cock the echanis for firing. 4. Start the software that controls the hotogate. 5. Fire the gun, aking sure that no one is in the ath of the all and that aroriate easure have een taken to sto the all after it hits the ground. 6. Record the velocity of the all 7. Reeat this rocedure 4 ties and average the velocities. Record these results on the activity sheet. Once we have the velocity of the all, we are ready to roceed to the second half of the exerient. Reove the hotogate and turn off the software efore roceeding. 1. Measure the ass (reove the two rass asses fro the otto of the endulu efore doing this) and the length of the endulu. Return the endulu to the device such that the all will e catured y the all when it leaves the end of the lunger.. Measure the ass of the all. Load the all into the launcher.

4 3. Making sure that the ath is clear, fire the all into the endulu. Measure the angle to which the endulu swings. Use this angle to deterine the height to which the endulu and the all went (H L (1 cos θ), where L is the length of the endulu). Record this data on the activity sheet. 4. Reeat this rocedure 4 ore ties. 5. Measure the ass of the two rass asses and record their su on the activity sheet. Attach the to the otto of the endulu. 6. Reeat stes Calculate the efficiency of the energy transfer and coare this to the theoretical value.

5 ESA1: Environental Science Activities Activity Sheet Ballistic Pendulu Nae: Mass of the all g Mass of the endulu g Velocity of all Velocity Run 1 Run Run 3 Run 4 Run 5 Avg. v i v i Angle Angle Run 1 Run Run 3 Run 4 Run 5 θ f (no ass) θ f (added ass) Height Angle Run 1 Run Run 3 Run 4 Run 5 Avg. h f h f (no ass) h f (added ass) No Added Mass Added Mass ( )gh f ex v i theory Percent difference 1. How close is the odel efficiency to the actual efficiency?. What are soe ossile characteristics of the exeriental aaratus that are not accounted for in the odel? Could these characteristics account for the differences etween easured and theoretical accelerations?