Conservation of Energy

Transcription

1 Cnservatn f Energy Equpment DataStud, ruler 2 meters lng, 6 n ruler, heavy duty bench clamp at crner f lab bench, 90 cm rd clamped vertcally t bench clamp, 2 duble clamps, 40 cm rd clamped hrzntally t 90 cm vertcal rd, brass sprng clamped t hrzntal rd 25 cm frm vertcal rd, 100 g mass, 0.5 kg mass wth 3 n cardbard square taped t bttm, mtn sensr II, phtgate sensr, nn-paper tube, paper tube abut 2.5 cm n dameter, bubble wrap, 30 cm strng wth lps at bth ends, calpers. Cmments: Check that 0.5 kg mass hangs s cardbard s hrzntal. If t s nt, carefully bend hk n mass s t s. The bench clamp screw hldng the vertcal rd shuld be perpendcular t the hrzntal rd hldng the sprng (t avd acustc reflectns frm the bench clamp screw). 1 Purpse The ttal energy E f a smple mechancal system s the sum f the ptental energy PE and the knetc energy KE. In the absence f frctn the ttal energy E f the system s a cnserved quantty s that E = KE + P E. In the absence f frctn, f the KE and PE change, they must change s that ther sum s equal t the ttal energy E. In ths experment several smple mechancal systems wll be examned fr ths prperty. 2 Thery The wrk-energy therem s btaned frm a spatal ntegratn f Newtn s Secnd Law. Let F be the frce n a pnt mass m whse pstn and velcty are r and v. If the mass mves frm an ntal pstn () t a fnal pstn (f) the wrk-energy thery states that F d r = ( 1 2 mv2 ) f ( 1 2 mv2 The left hand sde f ths equatn s defned as the wrk W and the rght hand sde as the change n the knetc energy KE. Fr the wrk, gven by an ntegral called a lne ntegral, there are 2 pssbltes. 1. The value f the wrk r lne ntegral depends n the path r rute taken by the mass as t mves frm () t (f). In ths case the frce s called nn-cnservatve. An example s the frce f frctn. A cnsequence f ths s that there s n functn whse dfferental equals the ntegrand f the lne ntegral and a defnte path fr the mass m must be specfed t evaluate the wrk. 2. The value f the wrk r lne ntegral des nt depend n the path r rute taken by the mass as t mves frm () t (f). In ths case the frce s called cnservatve. Examples are the frce exerted by a lnear sprng and the unfrm gravtatnal frce. In ths case there s a functn called U whse dfferental du s equal F d r. The functn U s called the ptental energy (PE). Fr case 2 the wrk-energy therem can nw be wrtten as ( ) ( ) 1 1 du = (U) f + (U) = 2 mv2 2 mv2 1 f ). = (KE) f (KE).

2 As U s evaluated nly at the pnts () and (f) t s clear that fr cnservatve frces the wrk depends nly n the end pnts and nt n the partcular path traversed by m. Ths equatn can be wrtten as (U) f + (KE) f = (U) + (KE). Each sde f ths equatn s called the ttal (mechancal) energy E f the mass. On the left E has been evaluated at pnt (f) and n the rght at pnt (). The quantty E has been cnserved and E f = E. Ths statement s called the cnservatn f energy. Energy s nt cnserved f frctn r ther nn-cnservatve frces are present. It s cnvenent t refer the PE at any pnt t a fxed reference pnt (). Fr a cnservatve frce the wrk ntegral frm () t (f) s ndependent f path. Let that path g thrugh the reference pnt (). The wrk ntegral becmes W = F d r = F d r + F d r = F d r + F d r = +U U f, where U = F d r and U f = F d r. U f s called the ptental at (f) relatve t () and U s called the ptental at () relatve t (). The cnservatn f energy statement may be wrtten U f + (KE) f = U + (KE). Frm ths last equatn t s evdent that any value can be gven t the PE at the reference pnt, fr ths value wll appear n bth sdes f the equatn and effectvely cancels ut. Fr smplcty, the value f the PE at the reference pnt s usually taken as zer and n what fllws we assume ths t be the case. The subscrpt referrng t the reference pnt s ften mtted but the PE s always wth respect t a reference pnt and the lcatn f the reference pnt shuld be clearly stated. In ne dmensn, f that dmensn s taken t be x, du = F dx, and 2.1 Lnear Sprng F = du dx. Let a sprng le n the x axs and assume ne end f the sprng s fxed. If x s the pstn f the free end f the unstretched lnear sprng, a mass attached t the sprng wll experence a restrng frce F = k(x x ) where x s the pstn f the free end f the stretched sprng, (x x ) s the length the sprng has been stretched and k s the sprng cnstant. Take the reference pnt fr the PE at x. The PE fr a mass attached t the sprng wll be U(x) = x x 0 F dx = x x k(x x )dx = 1 2 k(x x ) 2, where the ptental energy at x = x 0 has been taken as zer. Often, x s chsen as the rgn and U(x) = 1 2 kx2. 2

3 2.2 Unfrm Gravtatnal Feld Fr a mass m n the unfrm gravtatnal feld n the surface f the earth the frce f gravty s F = mg where g s the acceleratn f gravty and up s pstve. Take up as the pstve y crdnate and y as the PE reference heght. The PE at any heght y s gven by U(y) = y y 0 F dy = y y mgdy = mg(y y ), where the ptental energy at y = y 0 has been taken as zer. If y 0 s taken at the rgn f the crdnate, the PE becmes U = mgy. Ths PE s ften wrtten as mgh where h s the heght abve the reference pnt. (h s negatve f the mass s belw the reference pnt.) The PE depends nly n y and nt hw far the mass mves hrzntally. Hrzntal mtn des nt cntrbute t the PE. Why? 2.3 Multple Masses and Cnservatve Frces Fr a system cnsstng f a number f masses and frces, the analyss s easly extended. If all the frces, bth nternal and external, are cnservatve, the wrk dne by all the frces can be represented by ptental energes. The ttal energy s the sum f all the PE s and KE s f all the masses and ths quantty s cnserved. 3 Free Fall 3.1 Descrptn A tube wth mass m s held hrzntally a dstance h abve the level beam f a phtgate sensr. The tube s drpped (wthut rtatn) and ts velcty s measured as t passes thrugh the phtgate. The ttal energy at the tme f drppng s cmpared t the ttal energy as the tube passes thrugh the phtgate. 3.2 Thery Take the reference pnt fr the gravtatnal PE as the level f the phtgate beam. At the tme f drppng the ttal energy s mgh. If v s the velcty f the tube as t passes thrugh the phtgate the ttal energy at that tme s 1 2 mv2. Equatng the energy at the tme f drppng t the energy as the tube passes thrugh the phtgate, mgh = 1 2 mv2, whch gves v = 2gh. 3.3 Prgrammng Measure the dameter f the tube wth the calpers, checkng fr unfrmty. If the dameter f the tube s nt unfrm, dscuss ths n yur errr analyss. Prgram DataStud fr the dgtal phtgate sensr. In the experment setup wndw, g t Cnstants tab and enter the dameter f the tube where t says Flag Length. Open a dgts dsplay wndw fr velcty. Frm the drp dwn menu at the tp, ncrease precsn untl yu have 3 decmal dgts. 3.4 Takng Data Mve the phtgate s that ts plane s hrzntal and s ppste the base plate that hlds the vertcal rd supprtng the phtgate. Adjust the heght f the phtgate beam t be 25 cm abve 3

4 the table and place the phtgate ver the edge f the table next t the vertcal rd n the bench clamp. Drp the nn-paper tube s that t s hrzntal, des nt rtate, and s perpendcular t the phtgate beam. Use the vertcal rd next t the phtgate as a gude fr drppng the tube thrugh the phtgate wthut httng t. Clck START, and usng the meter stck drp the tube nt bubble wrap frm a heght h = 10 cm abve the beam, r a heght f 35 cm abve the table. Measure heght t the mddle f the tube. Recrd the velcty thrugh the phtgate. Yu wll want t take sme practce runs and a few runs fr real. Repeat fr h = 20 and 30 cm. 3.5 Analyss Cmpare yur measured velctes wth that predcted by the thery. Is there any frctn n ths experment? If s, hw wuld t affect yur data? Try drppng the paper tube frm a heght f 50 cm abve the phtgate beam and analyze the data as befre. Is energy cnserved? If nt, what happens t the energy? 4 Pendulum 4.1 Descrptn A 100 g mass s hung frm a 30 cm strng and used as a pendulum. The weght s pulled t ne sde and let g. At the lwest pnt the weght passes thrugh the beam f a phtgate sensr and ts velcty s measured. 4.2 Thery As the pendulum swngs dwn, PE s cnverted nt KE. The change n PE s gven by mgh where h s the vertcal dstance that the weght has mved. The weght travels the arc f a crcle, and ths dstance s lnger than h. The velcty s gven by exactly the same expressn as n the prevus experment, v = 2gh. The weght has a frce exerted n t by the strng, but ths frce des nt cntrbute t the PE. Why? 4.3 Prgrammng Measure the dameter f the 100 g weght wth calpers. Fllw the prcedures n the prevus sectn. 4.4 Takng Data Adjust the phtgate s that the legs pnt up and the beam s 10 cm abve the bench. The phtgate beam shuld be drectly belw the suspensn pnt f the pendulum. Adjust the heght f the suspensn pnt f the pendulum s that the mddle f the weght swngs thrugh the center f the phtgate beam. Pull the weght t ne sde s that the mddle f the weght s 10 cm abve the level f the beam. Let g and recrd the velcty f the weght at ts lwest pnt. Repeat fr h = 8, 6, and 4 cm. 4.5 Analyss Cmpare yur measured velctes t the theretcal values. Nte that t s nly the vertcal dstance thrugh whch the weght falls that cntrbutes t the change n PE. 4

5 5 Gravty + Sprng 5.1 Descrptn A vertcal sprng has a 0.5 kg mass m hangng frm t. The mass s set nt vertcal scllatry mtn and ts pstn s measured as a functn f tme by a mtn sensr. The velcty and acceleratn are calculated by DataStud. The ttal energy f the mass s cmpared at dfferent pnts n the mtn. The PE f the sprng-mass system s due t bth gravtatnal PE and the PE f the sprng. 5.2 Thery The mass m perfrms smple harmnc mtn wth an angular frequency gven by ω = k/m, where k s the sprng cnstant. Let y be the vertcal crdnate (up pstve) that gves the pstn f the end f the sprng frm whch m s hung. Take the crdnate rgn as the pstn f ths end f the sprng when there s n mass n the sprng (sprng unstretched). When the sprng s stretched dwnward by hangng a mass n t (y wll be negatve) the PE f the sprng wll be 1 2 ky2 where k s the frce cnstant f the sprng. Let the gravtatnal PE f the mass m be zer when y = 0 (sprng unstretched). The gravtatnal PE fr m s then mgy, whch wll be negatve. (Whle y s nt the crdnate f the mass, the end f the sprng and the mass mve the same amunt.) When the end f the sprng s at a pnt gven by y and has a velcty v = dy/dt (als the velcty f the mass), The ttal energy E f the sprng-mass system s gven by E = 1 2 mv ky2 + mgy. E shuld be the same at any pnt n the mtn, assumng n frctn. 5.3 Prgrammng Prgram DataStud fr the mtn sensr, usng the default values. Drag and drp the graph cn frm the dplays wndw dgtal nt the mtn sensr cn. The graph dsplay wll appear and yu shuld see the pstn labeled n the y axs. Next drag and drp the velcty and acceleratn cns frm the data wndw t the center f the graph dsplay, s nw yu shuld see pstn, velcty, and acceleratn n the vertcal axes. Yu wll prbably fnd the default samplng rate f 20 Hz t be satsfactry, but feel free t experment. 5.4 Prelmnares The sprng shuld be hangng ver the edge f the bench. Adjust the heght f the sprng s that wth the 0.5 kg mass attached, the bttm f the mass s abut 65 cm frm the flr. Wth the mass remved frm the sprng use the tw meter stck rented vertcally and the 6 n ruler rented hrzntally t determne the heght f the unclamped end f the sprng frm the flr. Attach the mass t the sprng and measure the heght f the same end f the sprng frm the flr. Determne k fr the sprng. Als, let the amunt the sprng stretches be y 0, a negatve number. Then, usng the crdnates already ntrduced, the pstn f the end f the sprng wth the mass attached and at rest s gven by y = y 0. Place the mtn sensr n the flr drectly beneath the the mass. Check that the face f the mtn sensr s hrzntal. There s a swtch n the mtn sensr labeled narrw r std (standard) whch sets the wdth f the acustc beam f the sensr. Yu wll 5

6 prbably fnd that std wrks the best but feel free t experment. The mtn sensr shuld be plugged nt the nterface s that pstn ncreases as the mass ges hgher. Set the mass nt mtn by pullng t dwn a reasnable amunt and lettng g. (the mtn sensr des nt recrd dstances less than 0.15 m frm ts face). D nt set the mass n mtn by lftng t up. If yu lft t t far t wll crash nt the mtn sensr. After the mtn settles dwn, clck START and d the fllwng. Smultaneusly bserve the mtn f the mass and the graphs f pstn, velcty, and acceleratn. Des yur ntutn abut the mtn crrespnd t what the graphs are dsplayng? Fr example, s the acceleratn maxmum r mnmum when the velcty s zer? When the velcty s maxmum s the acceleratn maxmum r mnmum? Observe the sprng-mass system. Is there knetc energy that s nt gven by 1 2 mv2? If s, ths wuld be wrth whle mentnng n yur errr analyss. (Hnt: There are at least tw tems ne mght ntce here.) 5.5 Experment and Analyss Set the mass nt mtn, and when the mtn settles dwn clck START, and clck STOP after 3 r 4 cycles f the mtn. Usng the Smart Tl n the pstn graph, determne the dstance the mass has traveled frm a chsen hghest pstn (H) t the fllwng lwest pstn (L) n the pstn graph. Ths dstance wll be 2A, where A s the ampltude f the mtn and s a pstve number. Determne the ttal energy E fr the fllwng 4 pstns f m durng the mtn: the hghest and lwest pstns f the mass utlzed n fndng the ampltude A, and the tw pstns f maxmum velcty fllwng these hghest and lwest pstns f the mass. Recall that the crdnate y gves the pstn f the end f the sprng, nt the pstn f the mass. The hghest pstn f the mass ccurs when y = y 0 + A and the lwest pstn f the mass when y = y 0 A. Maxmum velcty ccurs when y = y When the mass s at the hghest chsen pstn and the KE s zer, the energy E H wll be E H = 1 2 k(y 0 + A) 2 + mg(y 0 + A). (1) 2. When the mass has the maxmum dwn velcty fllwng the hghest chsen pstn f the mass the energy E 01 s gven E 01 = 1 2 mv ky2 0 + mgy 0. (2) 3. When the mass s at the lwest chsen pnt and the KE s zer, the energy s gven by E L = 1 2 k(y 0 A) 2 + mg(y 0 A). (3) 4. When the mass has the maxmum up velcty fllwng the lwest chsen pstn f the mass the energy E 02 s gven by E 02 = 1 2 mv ky2 0 + mgy 0. (4) Obtan the velcty frm the velcty curve f the graph dsplay. Cmpare the ttal energy at these 4 pnts. Is energy cnserved? What factrs mght ntrduce errr nt yur measurements and calculatns? 6

7 5.6 Questn 1. The analyss presented n ths sectn assumes n frctn. Hw wll frctn affect yur expermental results? D yu see evdence f frctn n the curve f pstn vs tme? 6 Fnshng Up Please return the bench t the cndtn n whch yu fund t. Frm the Fle menu clck New and then clck Dn t Save. Thank yu. 7