On Mopping: A Mahemaical Model for Mopping a Diry Floor Connor Palaucci McGill Universiy Faculy of Engineering
Absrac Several imes in my life I have been old ha mopping he floor is no a valid mehod of cleaning because i jus spreads he filh around. This is clearly false. As anyone who has mopped a floor can esify, he iniially clean waer in he mop bucke is sauraed wih dir by he end of he cleaning session. This dir sared on he floor and ended in he bucke, so clearly he floor has less dir on i afer mopping. Even afer his simple observaion, some criics of mopping remain unconvinced. The noion ha a cenuries-old janiorial echnique serves no purpose is as insuling o hose of he profession as i is false. Decriers of he mehod may be moivaed by a desire o appear worldly and experienced o heir peers, or perhaps hey may simply rying o avoid heir mopping duies. This paper does no aemp o provide insigh ino he moivaions of such naysayers, bu merely aims quell disbelief in he effeciveness of mopping by use of a mahemaical model. Cleaning mehod Model The following models he siuaion where an iniially clean mop and bucke of waer are used o clean a floor. Firs, he mop is weed wih he maximum amoun of waer i can hold. Nex, he mop is dragged across he floor, collecing dir. Dir coninues o collec in he waer on he mop unil he maximum sauraion of he mop waer is reached. The mop is hen dipped ino he waer in he bucke, deposiing all of is suspended dir and increasing he concenraion of dir in he bucke of waer. The mop is hen removed from he waer again, his ime however, he waer on he mop is no enirely clean. The mop waer has he same concenraion of dir as he bucke waer, and herefore can collec less ne dir from he floor before being sauraed again. This coninues unil he waer in he bucke is sauraed wih dir and he mop removed from he waer is also sauraed. Because he waer on he mop already holds he maximum possible amoun of dir, i can no longer remove dir from he floor. A his poin he waer and mop mus be cleaned. This process is repeaed unil a sufficien amoun of dir has been colleced from he floor.
Assumpions 1. The mop and bucke are iniially clean. 2. There is a maximum concenraion of dir ha waer can hold, by dragging he mop across he floor, his concenraion is reached in he mop. 3. The amoun of waer deposied on he floor by he mop is negligible. 4. All soluions are well-mixed. Le Q() represen he concenraion of dir in he waer bucke a any given ime. Noe ha he unis of Q() are grams per lier, and Q(0) = 0 as he waer is iniially clean. The volume of waer in he bucke is V B liers. Each ime he diry mop is reurned o he bucke, some amoun of dir, dd, is deposied in he bucke waer. This changes he concenraion of he bucke waer by he following equaion. d = dd The amoun of dir deposied, dd, depends on he maximum amoun of waer ha he mop can hold,. The waer on he mop sars wih an iniial concenraion of dir equal o ha of he waer in he bucke. Dir is hen added o he waer on he mop unil he waer is sauraed. If Q max represens he maximum concenraion of dir in waer, he following equaion shows ha he final sauraion of he mop comes from boh he iniial dir on he mop, Q(), and he dir picked up from he floor, dd. Rearranging Subsiuing his value ino he firs equaion Q max = Q() + dd dd = Q max Q() d = Q max Q() d + Q() = Q max
A firs-order linear differenial equaion which can be solved using he inegraing facor d e VM + μ = e Q()e = d d (Q()e VM ) = Q max e Q max e d (Q()e V M ) = Q V max e d T V V d (Q()e M M ) = Q V max e d T V M Q()e = Qmax e + C Q() = Q max + Ce Using he iniial condiion Giving he final funcion Q(0) = 0 0 = Q max + C C = Q max Q() = Q max Q max e
Dir Concenraion, Q, (g/l) Diriness of Mop Waer Over Time 100 90 80 70 60 50 40 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Moppings,, (ul) Figure 1. Plo of one soluion wih Q max as 100 g/l. Conclusions This model shows, as expeced, ha he marginal uiliy of mopping he floor decreases as he mop waer ges increasingly diry. However, he noion ha mopping he floor jus spreads he dir around is clearly false as each mopping moves dd grams of dir from he floor ino he bucke of waer. The arge audience of his paper is hose who, perhaps in an aemp o seem learned on he opic, decry mopping as an ineffecive mehod for cleaning floors, ye poses a basic undersanding of ordinary differenial equaions. The echniques presened here could be used in furher research o disprove he idea ha aking a bah is merely soaking in one s own filh.