Exploring the relationship between a fluid container s geometry and when it will balance on edge

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Clifford Simpson
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1 Exploring the relationship eteen a fluid container s geometry and hen it ill alance on edge Ryan J. Moriarty California Polytechnic State University

2 Contents 1 Rectangular container The first geometric regime The second geometric regime The single-slotted container Geometric regimes and meta-regimes Center of mass calculations Windo of alance Balancing indo reentry Illustrative center of mass trajectories Considering the container s mass Center of mass ehavior Illustrative center of mass trajectories incorporating the container s mass Conclusion 24 Appendix A Crossover h values 25 i

3 Introduction At some point hile consuming a everage, many people ill idly try to alance its container on edge. The act itself is physically straightforard, merely involving the system s center of mass and achieving a static equilirium eteen the opposing torques caused y gravity and the normal force eteen the container and the surface on hich it alances. Further analysis of the act, hoever, illuminates the richness of the exercise. These nuances are apparent even in simplified to-dimensional models ecause of the depth of the relationship eteen a container s geometry and achieving alance. The purpose of such analysis is threefold: first, hen considering a rectangular container, to determine the relationship eteen the angle at hich it alances and the amount of fluid in the container; second, to consider a massless analogue to a standard telve-ounce aluminum can hich alances at a fixed angle and oserve the interplay eteen the various parameters of that container s geometry and alance; and finally, to revisit the aluminum can model, this time considering its mass relative to the fluid s, and recover the familiar ehavior oserved hen alancing real-orld everages in aluminum cans. ii

1 Rectangular container Figure 1: To-dimensional rectangular container filled to height h ith liquid resting on its ase, idth, (left, and alancing on its edge (right.

4 1 Rectangular container Figure 1: To-dimensional rectangular container filled to height h ith liquid resting on its ase, idth, (left, and alancing on its edge (right. To facilitate this analysis, the fluid in the container ill e treated as uniformly dense. As such, calculating the center of mass simply requires calculating the centroid of a given fluid shape. There are to regimes for the geometry of the fluid hile its container is alanced on edge. The first to consider, hich exhiits the richest ehavior, is seen in Fig. (1. Later, the second regime here the fluid height in the tilted frame sits at or elo the elevated corner, ill e explored. It is orth nothing that Fig. (1 also illustrates the definition of alance. Its center of mass, laeled c.o.m., rests directly aove the corner of the container in contact ith the surface. By definition, for the container to e alanced, i.e., in mechanical equilirium, the net force and net torque on the system must oth e zero. Per Neton s Third La, the normal force and the eight of the fluid are equal in magnitude ut antiparallel, satisfying the first condition for mechanical equilirium. Because the container rests at an angle θ such that there is no perpendicular distance eteen either force vector and the corner of the container in contact ith the surface aout hich it could pivot, the net torque is zero, satisfying the second and final condition for mechanical equilirium. 1

5 Figure 2: Rectangular container alanced on edge in the first geometric regime. 1.1 The first geometric regime The variale is significant to the analysis of the rectangular container in this regime, ut its algeraic relationship ith the other knon parameters isn t immediately apparent. Setting the area of the fluid, h, equal to the sum of the areas of the to shapes in Fig. (2, i.e., h = + θ, 2 2 tan θ, it follos that, for a given untilted fluid height h and alancing angle = h 2 tan θ. (1 By taking a eighted average of the centroids of the triangle and rectangle hich constitute the fluid shape seen in Fig. (2, the x and y coordinates of the tilted fluid s center of mass are ( x, ȳ = 1 [ ( h 2, ( tan θ 3, + ]. (2 3 tan θ After sustituting Eq. (1 into Eq. (2, x and ȳ ecome x = h tan θ, (3 2

6 and ȳ = h h tan 2 θ. (4 As noted efore, hen the system is in static equilirium, the center of mass is directly aove the corner on hich the container alances. When such alance is achieved the x -coordinate of the fluid s center of mass on the rotated axis shon in Fig. (2 ill e zero, meaning that x = x sin θ ȳ cos θ = 0 (5 must e satisfied. Sustituting Eqs. (3 and (4 into Eq. (5, then multiplying through y h, it follos 2 that the quadratic ( h 2 tan θ ( h + 1 ( tan 2 θ + 1 = 0 (6 must also e satisfied for the container to achieve alance. From the ell-knon quadratic formula and Eq. (6, ( h as a function of θ in this regime is ( h = tan θ tan 2 θ + 1 ( tan 2 θ + 1. (7 While it ould e preferale to have found the inverse relationship, such a relationship cannot e found due to the transcendental nature of Eq. (7. 3

Figure 3: Rectangular container alanced on edge hen ( h 1 2. 1.2 The second geometric regime It is important to recall that Eq. (7 is only true in the geometric regime of Figs. (1 and (2.

7 Figure 3: Rectangular container alanced on edge hen ( h The second geometric regime It is important to recall that Eq. (7 is only true in the geometric regime of Figs. (1 and (2. That is, it only holds true so long as the tilted fluid height sits aove the far corner ( of the container. Restated in more concrete, algeraic terms, it is true only hile h 1 2, hich ill e apparent later. In the second geometric regime, shon in Fig. (3, the variale, B, has een introduced to facilitate this analysis. The center-of-mass calculation is more straightforard in the second regime, as the fluid is simply in the shape of a triangle ith the centroid located at ( x, ȳ = 1 h [ B 2 2 tan θ ( ] B 3, B. (8 3 tan θ Applying the rotational transform of Eq. (5 and recalling that the x -coordinate of the center of mass must once again fall at the origin, the condition required for alance to e achieved in this regime is B 3 tan θ = B tan θ. (9 3 There is only one angle, θ = π 4, for hich Eq. (9 holds true. Thus, the intuitive result 4

8 Figure 4: Graph of θ against ( h. for achieving alance in this regime has een recovered: the geometry of Fig. (3 yields a right triangle ith equal-length catheti. Beteen Eqs. (7 and (9, the massless rectangular container has een solved in each of the possile geometric regimes. In Fig. (4, the complete ehavior of the massless rectangular container is illustrated. In the limit here ( h On the other extreme, hen ( h, (that is, hen h >> the alancing angle θ approaches π 1 2, the alancing angle θ remains at π

2 The single-slotted container Figure 5: The single-slotted container alanced on its slotted edge In solving the rectangular container, one uncovers relationship eteen the properties of it and the

9 2 The single-slotted container Figure 5: The single-slotted container alanced on its slotted edge In solving the rectangular container, one uncovers relationship eteen the properties of it and the fluid it contains and the angle at hich it ill alance. Hoever, a common real-life experiment is to attempt to alance an aluminum can on the eveled edge hich surrounds its ase. In that prolem, the alancing angle is instead a fixed parameter of the container s geometry and the conditions for alance are a function of that geometry. Fig. (5 shos the to-dimensional model hich ill e used for analyzing the ehavior oserved hen attempting to alance aluminum cans on their evel. For alance, i.e. mechanical equilirium, to e achieved in this model, the system s center of mass must e directly aove the container s slot. If the center of mass is to the left of the slot, gravitational torque ill tip the container anticlockise onto its side. Similarly, if the center of mass is to the right of the slot, the torque ill tip it clockise ack onto its ase. For the rectangular container filled ith fluid height h, alance is alays achieved at some angle θ. Hoever, in this model, for a container of ase, slot idth and fixed alancing angle θ containing fluid of height h, alance is only achieved hen the center of mass of the system falls ithin the indo of the container s slot idth. Analysis of the single-slotted container to approximate the ehavior oserved hen alancing an aluminum can on its evel, then, is concerned ith the trajectory of the center of mass of the fluid in the tilted container as a function of the fluid s height hen the container is upright. 6

2.1 Geometric regimes and meta-regimes Figure 6: The four possile geometric regimes for the single-slotted container, I and II eing the untilted fluid geometries, and (a and ( the tilted fluid

10 2.1 Geometric regimes and meta-regimes Figure 6: The four possile geometric regimes for the single-slotted container, I and II eing the untilted fluid geometries, and (a and ( the tilted fluid geometries. As as done ith the rectangular container, the four fluid geometries shon in Fig. (6 ill e delineated y expressions involving the untilted fluid height h. To do this a schematic graph of the three meta-regimes of the fluid s geometry ill e utilized. Figure 7: Schematic detailing the three geometric meta-regimes of the single-slotted container. In Fig. (7, an Araic numeral indicates a unique meta-regime. A roman numeral comined ith a loercase letter (e.g., II indicates hich tilted and untilted geometric regimes from Fig. (6 are oserved in a particular container for a range of h values. For example, regime Ia corresponds to a container ith fluid height h hich has the untilted fluid geometry of diagram I in Fig. (6 and the tilted geometry of diagram (a also seen in Fig. (6. An h ith suscripts (e.g., h I,II indicates a crossover fluid height at hich a ne geometric regime - tilted, untilted or oth - occurs. For example, h Ia, indicates the h 7

11 value at hich fluid geometry transitions from regime Ia to I as h is decreased, and h I,IIa indicates the geometric transition eteen regimes Ia and IIa as h decreases, and so forth. These crossover heights are derived in Appendix A. Stated here ithout proof they are and h I,IIa = h I,II = h Ia,II = cos θ, (10 h Ia, = 2 tan θ, (11 h IIa, = ( sin θ + 2 ( sin θ. (12 tan θ With Eqs. (10, (11 and (12 the ehavior of the fluid geometry as a function of h is no knon ithin each of the three meta-regimes. Hoever, these crossover heights do not diagram the ay in hich a container s geometry determines hich meta-regime in hich it operates. The third meta-regime is a good starting point for this discussion, ecause the crossovers from I to II and from (a to ( from Figure (6 occur simultaneously for the same height, h = h Ia,II. Another condition for this simultaneity involves the parameter V, hich is diagramed in the next section. This condition, hich a container s geometry must satisfy to operate in meta-regime (3, is V = 0, (13 here the general form of V is V = 1 cos θ [ A + 1 ] 2 2 sin θ cos θ 2 sin θ, (14 hen h = h Ia,II, here A is the area of the fluid in the container. As seen in Fig. (6, the fluid area can e expressed in regimes I and II, respectively, as A I = h sin θ cos θ. (15 Comining Eqs. (10, (13, (14 and (15 yields A II = h( sin θ h2 tan θ (16 = 2 sin θ, (17 the relationship of the spacial parameters of the container hich must e satisfied for it to exhiit the ehavior of meta-regime (3. 8

12 In meta-regime (1, the fluid geometry transitions eteen tilted regimes (a and ( hile remaining in untilted regime I. It follos, then, that hen h = h Ia, the untilted fluid level must e aove the high corner of the slot, i.e., Sustituting Eq. (11 into the inequality of Eq. (18 yields h Ia, > cos θ. (18 > 2 cos θ, (19 the relationship of the spatial parameters of the container hich defines meta-regime (1. Containers hich occupy meta-regime (2 transition eteen untilted regimes I and II hile remaining in the tilted regime (a. By inspection of the geometry of tilted regime (a, the condition V > 0 (20 must hold true hen h = h I,IIa. It must also e noted that for the container to maintain its geometric shape its slot cannot extend past its unslotted corner, i.e., > sin θ. This comined ith Eqs. (10, (14 and (20 yields sin θ < < 2 sin θ, (21 the relationship of the container s spatial parameters hich defines meta-regime (2. Beteen Eqs. (19, (21 and (17, the affect of a specific container s dimensions on the ehavior its fluid geometry as a function of untilted fluid height is fully mapped out for all three meta-regimes detailed in Figure ( Center of mass calculations The goal of analyzing the single-slotted container is to determine the range of untilted fluid heights for hich a container of ase, slot idth and alancing angle θ ill alance. Because the real-life exercise of attempting to alance an aluminum can on its evel generally involves sutracting the fluid from an initially full can y drinking it, a good starting point for discussing the fluid s center of mass is tilted regime (a from Fig. (7, ecause it descries the tilted fluid geometry of any container for a sufficiently large untilted fluid height h. 9

13 Figure 8: Laeled geometry of tilted regime (a. Figure (8 details the geometry of tilted regime (a for the purpose of calculating its center of mass. To accomplish this, the green-shaded fluid (area A, red-shaded triangle ith hypotenuse (area sin θ cos θ, red-shaded triangle ith hypotenuse V (area 1 (+V sin θ2 2 V 2 sin θ cos θ and the large triangle composed of the three other shapes (area 2 tan θ ill e considered. The centroids of these shapes can e related to each other y the expression ( + V sin θ 2 ( + V sin θ, + V sin θ = A( x, ȳ 2 tan θ 3 3 tan θ + 1 ( sin θ 2 2 sin θ cos θ, cos θ ( 2 V 2 sin θ cos θ + V sin θ, V cos θ 3 3 here ( x, ȳ is once again the center of mass of the tilted fluid ith area A a = ( + V sin θ2 2 tan θ, ( sin θ cos θ 1 2 V 2 sin θ cos θ (23 It follos from Eq. (22, then, that the center of mass of the fluid in tilted regime (a 10

is [ ( x, ȳ = 1 ( + V sin θ 2 ( + V sin θ, + V sin θ A 2 tan θ 3 3 tan θ 1 ( sin θ 2 2 sin θ cos θ, cos θ 3 3 1 ( 2 V 2 sin θ cos θ + V sin θ, V cos θ ].

14 is [ ( x, ȳ = 1 ( + V sin θ 2 ( + V sin θ, + V sin θ A 2 tan θ 3 3 tan θ 1 ( sin θ 2 2 sin θ cos θ, cos θ ( 2 V 2 sin θ cos θ + V sin θ, V cos θ ]. ( Figure 9: Laeled geometry of tilted regime (. Figure (9 outlines the geometry of tilted regime ( for the purpose of determining the fluid s center of mass. To this end, the green-shaded fluid (area A, red-shaded triangle ith hypotenuse (area B sin θ cos θ and larger triangle ith ase B (area 2 2 tan θ ill e considered. Because the area of the triangle ith ase B is equal to the sum of the other to shapes areas, it follos that B = [ ( 2 tan θ A + 1 ] sin θ cos θ. (25 11

The three shapes centroids are related y the expression B 2 ( B 2 tan θ 3, B = A( x, ȳ + 1 ( sin θ 3 tan θ 2 2 sin θ cos θ, cos θ, (26 3 3 from hich it follos that the center of mass of the fluid in

15 The three shapes centroids are related y the expression B 2 ( B 2 tan θ 3, B = A( x, ȳ + 1 ( sin θ 3 tan θ 2 2 sin θ cos θ, cos θ, ( from hich it follos that the center of mass of the fluid in tilted regime ( is ( x, ȳ = 1 A [ B 2 2 tan θ ( B 3, B 1 ( sin θ 3 tan θ 2 2 sin θ cos θ, cos θ ] 3 3 (27 and its area is A = B2 2 tan θ sin θ cos θ, (28 here B is given y Eq. (25 and A is determined y either Eq. (15 or (16, depending on its untilted geometry. 2.3 Windo of alance Figure 10: A detailed diagram of the single-slotted container s indo of alance. No that the center of mass ( x, ȳ is expressed for oth tilted fluid geometries, the analysis of the single-slotted container returns to the idea of alance. To e in the indo of alance for this container means that the center of mass must lay aove or eteen the edges of the slot hen turned on its side. By inspection, the (x, y coordinates for these to edges are (0, cos θ and ( sin θ, 0. Applying the rotational transform from Eq. (5 to these oundaries as ell as to the coordinates of the fluid s center of mass yields cos 2 θ x sin θ ȳ cos θ sin 2 θ, (29 the mathematical expression of the container s indo of alance. 12

16 2.4 Balancing indo reentry For any container of ase, slot idth and alancing angle 0 < θ < π 2, here ( > sin θ, there is a range of untilted fluid heights for hich Eq. (29 is satisfied and alance is achieved. Hoever, one can imagine that there exist container geometries in hich the fluid s center of mass eventually leaves the alance indo for a range of h values, such that gravitational torque ill turn the container upright, i.e. x = x sin θ ȳ cos θ > sin 2 θ. (30 An inspection of the container s geometry reveals that for certain container geometries, as h decreases, reentry can occur in either regime (a or (. The center of mass, hoever, can only leave the alance indo in regime (a, ecause in regime ( the fluid s center of mass alays decreases along a line interpolated eteen the centroids of the to triangles shon in Figure (9, ( 1 3, 1 ( 3 tan θ and 1 2 sin θ, 1 2 cos θ. It follos, then, that if Eq. (30 is to e satisfied in regime (, it must e satisfied at the crossover eteen regimes (a and ( hen V = 0 and B =. From Eq. (28, the fluid s area at the turning point is A = 2 2 tan θ sin θ cos θ. (31 Sustituting Eqs. (31 and (27 along ith the turning point condition B = into Eq. ( (30, then rearranging the terms such that it is a third-order polynomial function of yields ( ( 3 ( 2 3 tan 2 sin θ + 1 θ 3 sin θ ( 1 + sin 2 θ > 0, (32 the inequality hich a container s geometry must satisfy for alancing indo reentry to occur in regime (. Sustituting the formula for the fluid s center of mass in regime (a from Eq. (24 into the condition for alancing indo reentry in Eq. (30 yields the quadratic inequality ith coefficients and γ = 1 6 αv 2 + βv + γ > 0, (33 α = 1 2 cos2 θ, (34 β = 1 ( 2 sin θ 1 1 tan 2 2 sin 2 θ, (35 θ ( 1 1 tan θ 6 sin θ ( 1 + sin 2 θ sin θ2. (36 13

17 The solution to the quadratic in the inequality of Eq. (33 is V = β ± β 2 4αγ. (37 2α V is alays positive in tilted regime (a. As a result, Eqs. (33 and (37 ecome β + β 2 4αγ 2α > 0 (38 and β β 2 + 4αγ > 0. (39 2α Eqs. (38 and (39 determine ho, if at all, reentry occurs in regime (a of a particular single-slotted container. If oth Eqs. (38 and (39 are satisfied, then the center of mass ill leave and susequently reenter the alancing indo in regime (a. If only Eq. (39 is satisfied, then the inequality of Eq. (32 ill also e satisfied and the center of mass ill leave the alancing indo in regime (a and then reenter in regime (. When neither Eq. (38 nor (39 is satisfied, then Eq. (32 ill also not e satisfied and the container ill not exhiit the phenomenon of reentry. ( Because only real V values are physical and useful to this analysis, the discriminant, β 2 4αγ, eing positive determines the oundaries of the inequalities in Eqs. (38 and (39. 14

Figure 11: ( versus θ ith different regimes for center-of-mass ehavior highlighted. Figure (11 is a graph of the curves resulting from setting the third-order polynomial in Eq.

18 Figure 11: ( versus θ ith different regimes for center-of-mass ehavior highlighted. Figure (11 is a graph of the curves resulting from setting the third-order polynomial in Eq. (32 equal to zero and from setting the discriminant from Eq. (37 equal to zero in a ( ( -versus-θ parameter space in the ranges 0 < < 10 and 0 < θ < π 2. The resulting graph shos reentry ehavior for all single-slotted container geometries. The hite area of the graph indicates hen < sin θ, the region here the singleslotted container doesn t maintain its geometry so it is ignored in this analysis. The red-shaded area indicates the range of container geometries for hich Eqs. (32, (38 and (39 are not satisfied and reentry does not occur. In the oundary eteen the red- and pink-shaded areas, Eq. (37 s discriminant is zero. In the pink-shaded region, the discriminant of Eq. (37 is positive, ut Eqs. (38, (39 and (32 are not satisfied so reentry doesn t occur. The oundary eteen the red- and lue-shaded areas, here the discriminant of Eq. (37 is zero, is the range of container geometries for hich the center-of-mass turning point occurs in regime (a, at the edge of the alancing indo, ut Eqs. (32 (38 and (39 are not satisfied and reentry does not occur. The lue-shaded area indicates the container geometries for hich the center of mass leaves and susequently reenters the alancing indo in regime (a. In this region, oth Eq. (38 and (39 are satisfied, ut Eq. (32 is not. 15

19 The oundary eteen the green and lue regions, here the polynomial in Eq. (32 equals zero, indicates the geometries for hich reentry occurs at the turning point eteen tilted regimes (a and ( hen B = and V = 0. Finally, the green-shaded region indicates containers for hich the center of mass leaves the alancing indo in regime (a and then reenters the indo in regime (. In this region, Eq. (38 is not satisfied, ut Eqs. (32 and (39 are. 2.5 Illustrative center of mass trajectories So far the single-slotted container s center of mass ehavior has een illustrated only in astract terms. To oserve the phenomena descried y the somehat esoteric analysis of the previous section, it is helpful to plot ( x, ȳ for a range of untilted fluid heights h ithin the corresponding container geometry. Figure 12: Center of mass trajectory for a container ith ( = 5 and θ = 50. Figure (12 shos a container hose geometry does not permit the center of mass of the fluid it contains to overshoot the alancing indo. As such, its geometry falls in the red area of the parameter space graph of Figure (11. Figure 13: Center of mass trajectory for a container ith ( = 4 and θ = 58. Figure (13 shos a container hose geometry is such that its fluid s center of mass overshoots and susequently reenters the alancing indo in regime (a. Consequently, 16

20 its geometry places it in the lue region of Figure (11. Figure 14: Center of mass trajectory for a container ith ( = 5 and θ = 60. Figure (14 illustrates a container ith geometric parameters such that its fluid s center of mass leaves the alancing indo in regime (a then reenters in regime (, putting it in the green region of Figure ( Considering the container s mass Because real-orld telve-ounce aluminum cans aren t massless, hen attempting to alance them on their evel one oserves that for a sufficiently small amount of fluid the can ill fall onto its side. This is ecause the x coordinate of the can s center of mass falls short of the alancing indo and its mass is sufficiently larger than the mass of the remaining fluid to ring the can-fluid system s total center of mass short of the alancing indo. To incorporate this ehavior into the model of the single-slotted container, the container s mass M ill e defined as M = c 2, (40 here c is an aritrary, variale constant used to determine the relative mass of the container compared to fluid of area 2. Because aluminum cans are more or less horizontally symmetric, the container s center of mass ill e located at ( X, Ȳ ( = 2, k, (41 here k is a constant used to determine ho far up the y-axis the container s center of mass is located. Using the familiar analysis of the previous sections, the center of mass of the containerfluid system can e expressed as [ 1 ( x tot, ȳ tot = A ( x, ȳ + 1 ( X, Ȳ ], (42 M + A M 17

21 here A and ( x, ȳ are determined y Eqs. (23 and (24 or Eqs. (28 and (27, depending on hich geometric regime applies for a given untilted fluid height h Center of mass ehavior A similar analysis to that of Section 2.4 can e utilized to determine the ehavior of the container-fluid system s total center of mass for a container of ase, slot idth, alancing angle θ and coefficients c and k. As efore, to determine the center of mass trajectory in tilted fluid regime (a s ehavior, one must look at hen the x component of the system s center of mass is at the edges of the alancing indo, i.e. hen and x tot = x tot sin θ ȳ tot cos θ = sin 2 θ (43 x tot = x tot sin θ ȳ tot cos θ = cos 2 θ. (44 Sustituting Eq. (42 into Eq. (43 yields a quadratic of V ith the form of Eq. (37 ith coefficients α 1 = 1 ( 2 cos θ, (45 ( ( 1 cos θ 2 ( β 1 = sin θ sin 2 θ, 2 2 tan θ (46 and and γ 1 = [ ( 1 2 sin θ c + 1 ( cos θ ck + c tan θ ( sin 2 θ c tan θ ] ( 1 6 tan 2 θ ( sin θ cos θ ( 1 + sin 2 θ. (47 Sustituting Eq. (42 into Eq. (44 yields a quadratic of V ith coefficients α 2 = 1 ( 2 cos θ, (48 ( ( 1 cos θ 2 ( β 2 = sin θ + cos 2 θ, 2 2 tan θ (49 γ 2 = [ ( 1 2 sin θ c + 1 ( cos θ ck + 3 tan θ ( + cos 2 θ c tan θ ] ( 1 6 tan 2 θ ( sin θ cos θ ( 1 + cos 2 θ. (50 18

Once again, the sign of the discriminant determines the sign of the solution to the quadratic, so the equations hich determine the system s center of mass ehavior in regime (a for a container of

22 Once again, the sign of the discriminant determines the sign of the solution to the quadratic, so the equations hich determine the system s center of mass ehavior in regime (a for a container of coefficients c and k in a ( -versus-θ parameter space are β 2 1 4α 1 γ 1 = 0 (51 and β 2 2 4α 2 γ 2 = 0. (52 Figure 15: Parameter-space graph for a container ith center of mass coefficients c = 0.1 and k = 0.7 highlighting its center of mass ehavior in tilted fluid geometry regime (a. Figure (15 is the result of plotting Eqs. (51, (52 and ( = sin θ for a container ith coefficients c = 0.1 and k = 0.7. The red-shaded portion of the graph indicates container geometries for hich the center of mass never enters the alancing indo in regime (a. The green-shaded portion indicates geometries for hich the center of mass enters the indo of alance in (a, ut does not leave it in (a. The pink-shaded region indicates geometries for hich the center of mass enters the alancing indo in (a and susequently overshoots the indo in (a. The hite portion, as efore, indicates the region in hich < sin θ here the container does not maintain its geometry and as a result isn t part of this analysis. Figure (15, hoever, says nothing aout the ay the container s center of mass ehaves in regime (. To determine the ehavior of the center of mass in regime (, the values of B hich satisfy Eqs. (43 and (44 must e analyzed. Comining Eqs. (27, (42, (43 19

and (44 yield the polynomials and ( ( 1 3 ( sin θ sin θc cos θck + 2 sin 2 θc ( ( ( 1 3 ( sin θ sin θc cos θck + 2 + cos 2 θc ( 6 tan θ cos θ ( B 6 tan 2 θ 2 ( sin2 θ B 2 tan θ 6 tan θ cos θ ( B 6

23 and (44 yield the polynomials and ( ( 1 3 ( sin θ sin θc cos θck + 2 sin 2 θc ( ( ( 1 3 ( sin θ sin θc cos θck cos 2 θc ( 6 tan θ cos θ ( B 6 tan 2 θ 2 ( sin2 θ B 2 tan θ 6 tan θ cos θ ( B 6 tan 2 θ 2 ( + cos2 θ B 2 tan θ sin θ cos θ ( 1 + sin 2 θ = 0 ( sin θ cos θ ( 1 + cos 2 θ = 0. (54 Because Eqs. (53 and (54 are three dimensional polynomials, the center of mass ehavior for a particular container ith mass coefficients c and k must e analyzed for individual, fixed ( values for the range of ( B values 1 ( B (. Figure 16: Parameter-space graph for a container ith center of mass coefficients c = 0.1, k = 0.7 and ase-to-slot-idth ratio ( = 3.3 highlighting its center of mass ehavior in tilted fluid geometry regime (. 20

24 Figure (16 shos one such ( B -versus-θ graph for a container ith coefficients c = 0.1 and k = 0.7 for a fixed ( ratio of 3.3. The red-shaded region indicates θ values for hich the center of mass is alays past the alancing indo in regime (. The purpleshaded area indicates θ values at hich the center of mass reenters the alancing indo and susequently overshoots it in regime (. The green portion of the graph corresponds to angles for hich the center of mass reenters the indo and remains there in regime (. The lue-shaded area is associated ith θ values for hich the center of mass is alays inside of the alancing indo in geometric regime (. The pink-shaded region corresponds to angles for hich the center of mass leaves the indo in regime ( and remain short of it. The urgundy-shaded region is associated ith values of θ for hich the center of mass enters the alancing indo in (, leaves it in ( and remains short of it. Finally, the yello shaded region denotes angles for hich the center of mass is alays short of the alancing indo in regime (. With parameter-space graphs like Figures (15 and (16, a general outline of the centerof-mass ehavior for a particular single-slotted container ith ase, slot idth, alancing angle θ and container mass coefficients c and k can e predicted Illustrative center of mass trajectories incorporating the container s mass To illustrate the center of mass ehavior descried astractly in Figures (15 and (16, the system s center of mass, x tot, ill e plotted inside of the container ith y values equal to the corresponding untilted fluid height h. This is ecause as h gets loer, the system s center of mass trajectory doules ack on itself, eventually converging on the container s center of mass, ( X, Ȳ. As a result, plotting ( x tot, ȳ tot makes the trajectory unnecessarily difficult to parse. Fortunately, ȳ tot does not contriute to hether or not the container ill alance so its omission does not result in a loss of any information meaningful to this analysis. In these graphs, the green curve represents ( x tot, h. The lue curve corresponds to ( x, h, shoing the fluid s center of mass. The red X shos the location of the container s center of mass. The dashed lack lines denote the alance indo. Finally, the red lines sho the edges of the single-slotted container eing analyzed. 21

25 Figure 17: Center of mass trajectory for ( = 3.3, c = 0.1, k = 0.7 and θ = Figure (17 shos a container for hich its center of mass sits just past the alancing indo. As a result, the system s total center of mass overshoots the indo in regime (a, then reenters in regime ( and overshoots the indo again in regime (. The container s geometry and its center of mass ehavior correspond to the pink-shaded region of Figure (15 and the purple-shaded region of Figure (16. Figure 18: Center of mass trajectory for ( = 3.3, c = 0.1, k = 0.7 and θ = Figure (18 shos a container hose center of mass center of mass is just inside the right side of the alancing indo. As such, the container-fluid system s center of mass overshoots the indo in regime (a, then reenters the alancing indo in regime ( and remains there. The container s geometry and its center of mass ehavior correspond to the pink-shaded region of Figure (15 and the green-shaded region of Figure (16. 22

26 Figure 19: Center of mass trajectory for ( = 3.3, c = 0.1, k = 0.7 and θ = Figure (19 illustrates the center of mass trajectory for a container hose center of mass lay firmly inside of the alancing indo. The system s center of mass enters the alancing indo in regime (a, then never leaves it in either regime. This ehavior of indicative of the green-shaded region of Figure (15 and the lue-shaded region of Figure (16. Figure 20: Center of mass trajectory for ( = 3.3, c = 0.1, k = 0.7 and θ = Figure (20 shos the total center of mass ehavior for a container hose center of mass sits just outside the indo of alance. The center of mass of the container-fluid system enters the alancing indo in regime ( and susequently leaves it to fall short of the indo, also in regime (. The container s geometry and the system s center-of-mass trajectory correspond to the red-shaded area of Figure (15 and the urgundy region of Figure (16. Worth noting is that the ehavior seen in Figure (20, here alance is achieved for a range of h values ut otherise the system s center of mass falls short of the indo, is the ehavior oserved hen attempting to alance a real-life telve-ounce aluminum can on its eveled edge. Thus, the familiar ehavior of real-orld cans has een recovered y the to-dimensional, single-slotted model. 23

27 Figure 21: Center of mass trajectory for ( = 3.3, c = 0.1, k = 0.7 and θ = Figure (21 shos the center of mass ehavior of a container hose on center of mass falls relatively far short of the alancing indo. The system s center of mass never reaches the indo of alance. This ehavior corresponds to the red-shaded region of Figure (15 and the yello-shaded region of Figure (16. 3 Conclusion Three to-dimensional models of fluid containers have een examined. The analysis of these containers has fully mapped out the interplay eteen the models physical parameters and achieving alance. For the massless rectangular container, alance can alays e achieved. Its alancing angle is determined y y a function, Eq. (7, of the ratio of its ase to the untilted fluid height in the container. Similarly, for the massless single-slotted container hich alances at a fixed angle, θ, alance can alays e achieved for ranges of untilted fluid heights specific to the geometry of the container. The ehavior of the fluid s center of mass is determined y the inequalities of Eqs. (32, (38 and (39. These inequalities are mapped out in the ( -versus-θ parameterspace graph of Figure (11, hich illustrates the the ehavior of the fluid s center of mass in the container. Finally, the single-slotted container s on mass as taken into consideration to give a more accurate model of alancing a real-orld telve-ounce aluminum can on its eveled edge. Parameter-space graphs like Figures (15 and (16 can e used to fully map out the fluid-container system s center of mass ehavior for any container. It as also demonstrated that this model can produce the ehavior oserved hen alancing telve-ounce cans. Despite using a simplified, to-dimensional model of an aluminum can, the nuances of the ehavior oserved in the real orld have een recreated. An analysis of this model has shon that the depth of the prolem is not lost in the simplicity of the model. 24

28 Appendix A Crossover h values From Figure (6, it s clear that any transition eteen untilted regimes I and II occurs hen the fluid is at the top edge of the slot. These crossover heights, then, are h I,IIa = h I,II = h Ia,II = cos θ. (55 From Figure (6, the transition eteen tilted regimes (a and ( clearly occurs hen the level of the tilted fluid is at the corner of the ase of the can. The area at this transition is expressed as A a, = 2 2 tan θ sin θ cos θ. (56 Setting Eq. (15, the area of fluid in untilted regime I, equal to Eq. (56 and solving for h yields the next crossover height, h Ia, = 2 tan θ. (57 Finally, setting Eq. (16, the area of fluid in untilted regime II, equal to Eq. (56 yields the quadratic ith solutions 1 2 tan θh2 + ( sin θ h sin θ cos θ 2 2 tan θ = 0, (58 h IIa, = ( sin θ ± 2 ( sin θ. (59 tan θ Because only positive values of h are physical, the negative solution to the quadratic is rejected, yielding the final crossover height, h IIa, = ( sin θ + 2 ( sin θ. (60 tan θ 25

Logic Effort Revisited

Logic Effort Revisited Mark This note ill take another look at logical effort, first revieing the asic idea ehind logical effort, and then looking at some of the more sutle issues in sizing transistors..0