Supplement to Fun wit oueold object and centre of ma, originally publied in te German journal Pyik in unerer Zeit, 9-96 eflection on te drinking bowl Balance I te bowl made of maive tainle teel? No: ma of bowl m 5g tere are alo verion wit ligtly different mae. Te bowl diplace a volume of 75cm. Terefore, te denity would be ρ m/ gcm -. Stainle teel, owever, a 7.9 to 8.gcm - Additional quetion: Were would te centre of ma be if te bowl were really made of maive tainle teel? Ti would be very expenive and alo not function in te way intended! To wic ide would te bowl incline and ow muc? Te anwer i omewat more complicated! but poible wit te calculation under point 5 Wat i te tickne of te tainle teel eet under te precondition tat all te part of te eet ave a uniform tickne and tat te part are perical cap? Surface of a perical cap wit te eigt : O Te meaurement were made wit a liding calliper and are own in te figure on te rigt. Tere are error in meaurement of about ±.5mm. Wit te vector-orientated program CorelDraw, te inner circle wa drawn troug te inner bottom pole and te two inner upper border point. A calculation by feet would be relatively exauting. For te urface of te outer perical cap: O.9 5..cm urface of te inner perical cap: O.57.9.6cm Te upper eccentric circular ring a a urface of O.7 cm. cm.8cm Te um of tee tree urface part i O 57.5cm Wit 57.5cm d 7.9gcm - 5g, te tickne d of te tainle teel eet can be calculated under te precondition tat te tickne i uniform all over: d.75cm
Supplement to Fun wit oueold object and centre of ma, originally publied in te German journal Pyik in unerer Zeit, 9-96 Were i te centre of ma of te entire empty - bowl under te precondition tat te tickne of te eet i uniform all over? Can te centre of ma be were you calculate it under ti precondition? Centre of ma of a tin walled perical cap ti formula can be found in te internet: - / e.g. for te centre of ma of te outer perical cap -9.-5./ -.5mm mean i located.5mm under M ditto -.-5./ -9.mm under M Te centre of ma of te eccentric circular ring can be calculated from -x m - x m /m - m -t/d /d - -/7/6 - -7.mm wit t x x mm, and x, and m r dρ ditto m Tee tree centre of ma are marked in te upper drawing. From and you can calculate te common centre of ma of bot of tee lever rule. From ti and you can calculate te centre of ma of te entire empty bowl marked wit a green cro. Ti centre of ma i on te rigt ide in regard to te connection line between M and A. Terefore, te empty bowl ould incline to te rigt. Ti i obviouly not te cae. Ti centre of ma mut be on te line at from M. Only ten would te empty bowl incline to te left under ti angle. Te implet metod to correct ti would be to add an additional ma G on te left. Ti ma G could be idden in te ollow part between te two perical cap. Oter poibilitie are tat te tickne of te tainle teel eet i not uniform all over or tat te ollow pace between te perical cap i filled wit a liquid, and or oter material. Perap tere are even more poibilitie.
Supplement to Fun wit oueold object and centre of ma, originally publied in te German journal Pyik in unerer Zeit, 9-96 How can you determine te additional ma G under te border of te bowl witout detroying te bowl? Aumption: te additional ma G i on te left under te ticker border of te bowl. Since te complete ma of te bowl i fixed, te tickne of te eet mut decreae. G mut be o great enoug to ift te centre of ma of te complete bowl on te line from M at blue cro. On te one and, te condition mut exit tat wit d a te unknown new tickne of te eet G 57.5cm 7.9gcm - d 5g On te oter and, te ratio between l /l mut conform wit te ratio of te ma G to te complete new ma calculated wit te new tickne d lever rule; determination of centre of ma l /l G/57.5cm 7.9gcm - d From te drawing, you can get te ratio l /l /.5. From equation and, d.67cm.67mm and G 8 g. Te tickne of te eet decreae a little from.75mm to.67mm. Te additional ma ift te centre of ma into te rigt poition. Wit te meaurement error already mentioned, te uncertainty of te calculation i etimated at about %. I actually cut open a Balance bowl ee figure on rigt. A you can ee, te aumption of an additional ma i verified. A roug etimation of tee two tainle teel mae ρ 8gcm - come to about g part :.5mm x6mm x5mm, part :.5mm x.5mm x5mm Ti conform rougly to te calculation. Te additional mae eem to be welded firmly to te eet. Tey cannot be removed witout force. Te tickne of te eet i about.7mm. Te bowl i probably produced in a deep-drawing proce te company carl merten doe not make production detail available.
Supplement to Fun wit oueold object and centre of ma, originally publied in te German journal Pyik in unerer Zeit, 9-96 5 Calculation of te volume and te centre of ma of a perical egment perical egment maive perical cap olume and centre of ma of a perical egment Te formula derived for ti are not eaily found explicitly on te internet or in matematical formulary. Te component extend from to ; te oriontal cut ave a radiu dependent on, namely becaue ti deal wit full circle wit between to. Terefore, te perical egment in cylindrical coordinate a te following repreentation: Spere:,, r olume egment r dr d d d d d wit follow [ ]
Supplement to Fun wit oueold object and centre of ma, originally publied in te German journal Pyik in unerer Zeit, 9-96 5 centre of ma d dr d d r d d d d Tet: Wit, te correct calculation i tu Wit, te correct calculation i tu -/8 ti i jut a emipere Wit /, te correct calculation i tu -/8 Wit, tere i a impler formula: [ ] [ ] Te reference point i te centre of te pere.
Supplement to Fun wit oueold object and centre of ma, originally publied in te German journal Pyik in unerer Zeit, 9-96 6 Determination of te centre of ma of te bowl wit liquid water/wine How i te centre of ma influenced by te level of te liquid? I te centre of ma directly above te bottom pole A? Te bowl i filled wit water to a level of mm, i.e. about 5mm below te edge, wic i quite full. Wine and water ave almot te ame denity. Te volume of te liquid i calculated wit te formula from point 5 cm g of water Te centre of ma i alo calculated wit te formula from point 5-8,mm Ti centre of ma i perpendicular below M. Wit te ditance of te centre of ma of te bowl blue cro and te liquid red cro and te related mae, te total centre of ma green cro can be calculated. Te ditance can be evolved from te drawing. Ideally, ti total centre of ma ould be located on te line between M and A. Ti i not exactly true ere, again probably due to error in meaurement. If more liquid i filled into te bowl, te total centre of ma would ift a bit more to te rigt and tu reult in even better tability. Te bowl tand uprigt even wit different level of liquid. How i ti poible? Accurate inpection of te bottom pole of te bowl ow tat tere i a flattening of about.mm ee potograp. Te black line in te potograp i a circle wit te diameter of te bowl. Ti mean tat tere i a mall, almot flat bae of about one centimetre. Terefore, mall variation of te total centre of ma can be compenated. 6