AB # - Equilibrium (saic) Inroducion Isaac Newon's conribuion o physics was o recognize ha despie he seeming compleiy of he Unierse, he moion of is pars is guided by surprisingly simple aws. Newon's inspiraion was o separae he moion of a paricle from he forces ha cause CHANGES in ha moion. This is saed in Newon's Second aw of Moion, ha: () d r() d NET ma() m m d () d where NET is he oal (ecor sum) of all he forces eered on a paricle wih mass m, posiion r(), elociy (), and acceleraion a(). Tha is, Eq. is a differenial equaion ha can be used (along wih he iniial condiions - he iniial posiion and elociy) o deermine he posiion and elociy of he paricle in quesion for all imes following. (This led Pierre Simone de aplace o claim ha a hypoheical, omniscien being ha knew he posiion and elociy of eery paricle in he Unierse and could sole Eq. would know he enire hisory of eery paricle in he Unierse, pas and fuure. I also brings up some annoying poins abou fae and free will; bu I digress). Mos paricularly ineresing applicaions of Eq. are hose in which here is some force acceleraing he paricle in quesion (he force of graiy ha is responsible for he moion of he planes and galaies and so forh). Howeer, a ery imporan area of engineering is he area of saics, in which i is he goal o se hings up so here isn' any acceleraion (ask he people of Souhern California how hey feel abou acceleraing buildings...). This condiion, in which here is no acceleraion, is called equilibrium. Referring o Eq., i should be clear ha if here is no acceleraion (i.e., if a() ), here can be no ne force acing. Noe ha his doesn' mean here are NO forces acing - i is simply he case ha he ecor sum of all forces acing on he paricle is zero. If we consider Eq. under condiions of equilibrium, we hae: () d NET m () d from which i is sraighforward o see ha he elociy is a consan independen of ime. Inegraing his equaion o ge he posiion r(), we ge () r r + (3) where r is he posiion a ime and is he elociy a ime. Recall ha Eq. 3 is really hree equaions, relaing he hree componens of he ecors. Tha is, Eq. 3 is shorhand for: y z () +, () y + y, () z + z, which can be rearranged ino he form: () y() y z(), y, z z, This can be recognized as he equaion for a sraigh line. Newon made his saemen in his irs aw of Moion; ha a body no aced upon by a ne force will moe in a sraigh line. Howeer, we are ineresed in a more resriced case, he case in which he force is zero AND he iniial elociy is zero (people in Souhern California would be jus as unhappy if heir houses moed in a sraigh line - i's he moing ha is a problem). If, hen Eqns. 4 simply say ha, y, and z neer change from heir iniial alues. (4) (5)
Tha is, he paricle in quesion neer moes. This is wha is called saic equilibrium. An eample of equilibrium would be a plane in leel fligh. The four forces acing on i are weigh (down), hrus (forward), drag (backward) and lif (up). In his case, he equilibrium is no saic (hopefully). An eample of saic equilibrium migh be a book on a able being pushed, bu no enough o oercome saic fricion. In his case, he four forces are he push (lef), saic fricion (righ), weigh (down) and normal force from he able (up). In hese wo eamples, he forces are already a righ angles o each oher. In general, his is no he case. This lab is inended o consider cases in which he forces in quesion are a angles oher han 9 o one anoher and sill add o zero. In order o se up such a siuaion, we use wha is called a force able. This is a circle wih angle in degrees marked around he perimeer, and pulleys ha can be clamped ino place a any angle. In his way, he force due o a weigh suspended from he pulley can be redireced ino any horizonal direcion desired. In wha follows, you will be gien Specific forces o se a specific angles, and you will be gien he ask of deermining he force (magniude AND direcion) necessary o balance he forces gien. This will be done graphically (in which you will represen he forces as ecors drawn on paper) and compuaionally (using sandard ecor analysis echniques). Eample Graphical sum As a simple eample, consider he case of a 5 N force a an angle of and anoher 5 N force a 9. These wo forces can be drawn on a piece of paper as: using a scale of cm of lengh represens N of force (in general, you should use a scale ha makes he drawing as large as possible o increase accuracy). The firs sep in deermining he force necessary o balance hese wo forces is o find he ecor sum of hese wo forces. The graphical mehod o add wo ecors is o shif one of hem (eiher one - he choice is arbirary) so ha he wo ecors are nose o ail. Noe ha he shif mus be accomplished WITHOUT changing he ecor's direcion OR magniude. This would look like he following:
The ecor sum is hen formed by joining he ail of he firs ecor o he nose of he las one: The las sep in deermining he ecor sum is o measure he lengh of he ecor so formed o find he magniude of he ecor sum, and he angle i presens o he -ais o deermine he direcion. The former ask is accomplished wih a ruler and he conersion scale (e.g., cm N) and he laer wih a proracor. (in his case, he lengh would correspond o a force of abou 7 N and he angle would be near 45 ). The final sep is o recognize ha he force necessary o balance his ecor sum is a force of equal magniude bu opposie direcion. How do you go in he opposie direcion? You "urn around 8 ". Tha is, add 8 o he angle deermined for he ecor sum (in his case, 8 + 45 5). Compuaional sum Using he sandard Caresian coordinaes, hese wo forces can be wrien as: + 5ˆ N + 5ˆ yn (6) The ecor sum of hese wo forces can be wrien as: ( 5ˆ 5yˆ )N (7) + Using he Pyhagorean heorem and he definiion of he angen, i is sraighforward o show ha he magniude of his equialen force is 7.7 N a a direcion of 45. We wish o find a SINGE force,, ha will balance. Tha is we seek such ha: + (8)
Clearly, - is he force we need. Tha is, if we suspend a 5 N force from he mark and a 5 N force from he 9 mark, hey will be balanced by suspending a force of ( 5ˆ 5yˆ )N (9) from he force able. The problem is ha Eq. 8 gies in componen form, no as single force wih a direcion (i.e., how do we suspend a SINGE force of (-5-5y)N from he force able). To accomplish our goal (balancing wih a single force), we mus deermine he magniude and direcion of. This is as sraighforward as before. The magniude is found using he Pyhagorean heorem: ( 5) + ( 5) N 7. 7N () and he direcion using he definiion of angen: y 5N an + 5N I simply punch his up on my calculaor and i ells me o direc he 7.7 N force a a direcion of...45. A lile hough, howeer, will make i clear he direcing halfway in beween he wo forces consiuing will NOT balance hem. So wha could possibly be wrong? This is one of hose rare imes when you hae o be smarer han your calculaor is. When you ask your calculaor which angle has a angen of one, i only gies you one of he answers (i.e., for each angle in he firs quadran, here is an angle in he hird quadran ha has he same angen - dio for he second and fourh). You simply hae o realize ha and accommodae. In his case, a ecor wih boh componens negaie is locaed in he hird quadran. The angle in he hird quadran wih a angen of one is 5. Thus, in order o balance a force of 5 N a and 5 N a 9, a force of 7.7 N mus be applied a an angle of 5. This can be erified on he force able. Error Analysis In he graphical porion, error analysis can mos sraighforwardly be accomplished by aking an aerage and sandard deiaion of he resul from each member of he group. Alhough he number of samples will be small, he error will be a leas roughly indicaed by his procedure. Error analysis for he compuaional par can be accomplished using he law of sines and of ines. Gien wo arbirary ecors of lengh and, and wih angle beween hem (measured from ), he lengh of he ecor sum is gien by R + + () R will, in general, hae hree sources of error due o, and. You can derie he following formula for he error in he magniude: R ( + ) σ ( + ) σ sin σ σ (3) One hing o recognize is ha unlike many preious cases, he error due o any one measured quaniy canno be cleanly separaed from and he oher measured quaniies. Noe also ha here are compensaing effecs. or eample, if is near (or 8 ), he error due o will be minimized bu he error due o and will be accenuaed. The aw of Sines can be used o deermine he direcion of he resulan (i.e., angle Φ beween and he resulan): sin Φ sin R wih a corresponding epression for he error in Φ: () (4)
σ sin sin + Φ σ σ σ R Φ Φ Φ Noe ha raher han subsiuing he epression for R, we simply use he calculaed error in R preiously calculaed. Your mission: You are asked o perform he same funcion using forces ha will be assigned o you. In he firs par, deermine he force ha mus be applied o balance he wo forces proided o you by your insrucor. In he second par, you will be gien THREE forces and you mus deermine he single force necessary o balance all hree. In each case (wo and hree forces) you mus deermine he answer boh graphically and compuaionally, and es your answer by seing up he assigned forces and your compued one on he force able. Your insrucor will hen come around a your iniaion and check o see if your soluion works. inally, check he error by arying he angle and magniude of he forces on he able and deermining how much of a change is necessary o hrow he sysem ou of balance and compare his o he error you calculae. (5)