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The Weight of Time Ian H Note The Weight of Time Ian H. Redmount, Department of Science and Mathematics, Parks College of Engineering and Aviation, Saint Louis University, St. Louis, MO 63156; and Richard H. Price, Department of Physics, University of Utah, Salt Lake City, UT 84112 s physics teachers, we are impulse is equal to the always on the lookout for net change in momen- Aproblems that can illustrate tum of the system dur- the sometimes subtle implications of ing the time interval. physical laws, particularly problems We can apply this to the that can be concisely stated and easi- interval that starts just ly visualized. One such problem is before the sand begins remarkable for the way it illustrates to flow, and ends just several fundamental principles of after it stops flowing. At mechanics. It is a wonderful problem both these endpoints the for stimulating discussion, even system has no momen- among students who have had only tum, so the change in an introduction to mechanics. momentum—and thus Fig. 1. Three types of hourglasses for which calculations are Suppose we have an old-fashioned the impulse—are zero. done: (a) cylindrical “egg timer” type, (b) spherical vessel type, hourglass, like any of those pictured in This means that the and (c) conical hourglass. Fig. 1, and suppose that it has been sit- average force on the hourglass-and- tem is moving downward at constant ting inactive on a perfect and perfectly sand system is zero. There are only velocity, then there is no net force sensitive scale which reads exactly two forces acting on the system. One and the apparent weight (the scale “10 newtons” with the hourglass on it. is the downward force of gravity, reading) must be exactly 10 N. But if We now turn the hourglass over and which we know to be exactly 10 N the center of mass is accelerating instantly put it back on the scale. For downward. The other force is the downward (upward) the apparent the next hour, while the sand is run- force of support by the scale. This is weight will be less (greater) than 10 ning, we observe the “apparent also what the dial on the scale indi- N. weight,” that is, the scale reading. Do cates, and what we call the apparent That acceleration is remarkably we see a reading of exactly 10 N? weight. From impulse considerations easy to calculate, using some reason- More? Less? On the one hand, since a we can conclude that the average of able simplifying assumptions. At portion of the sand in the glass is in this apparent weight must be 10 N times not too close to the start of the free fall, it seems that the hourglass upward. If at some stage during the sand flow, or to its conclusion, the should weigh less. On the other hand, flow of the sand the scale registers flow may be approximated as a uni- the impact of the sand on the base of more than 10 N, then there must be form (in volume per unit time) emp- the hourglass should increase the another time when the reading is less tying of the upper vessel of the hour- downward force exerted on the scale. than 10 N, and such discrepancies glass and filling of the lower vessel. Which of these effects is greater? Is must average to zero. Since the sand is particulate, rather either of measurable significance, or than a Newtonian fluid, the flow rate are both imperceptibly small? Do any A Matter of Acceleration is assumed to be a constant V/T, other effects contribute? It turns out Another principle of mechanics where V is the total volume of sand that this problem admits fairly can give us more details about when and T the time interval measured by straightforward analyses. discrepancies from the average 10 N the glass. (The flow rate for a apparent weight occur. We know that Newtonian fluid, such as water, A Matter of Impulse the acceleration of the center of mass would vary with the pressure head Impulse is the product of a time of a system is equal to the net force provided by the fluid in the upper interval and the average force exerted on the system divided by the mass of vessel and would decrease with on a system during that interval. the system. If the center of mass of time.) If the geometry of the glass According to the laws of mechanics, the hourglass-plus-flowing-sand sys- and sand is parametrized as in Fig. 1, 432 THE PHYSICS TEACHER Vol. 36, Oct. 1998 The Weight of Time A conical hourglass, in which each with y1 the level of the sand above the duration of the flow, except at the base of the lower vessel, y the level very beginning. vessel has height b and base radius 2 ␲ 2 of the sand above the throat in the This upward acceleration of the R, has Al(yl) = [R(b – y1)/b] and ␲ 2 upper vessel, and b the height of the sand’s center of mass would engen- A2(y2) = (Ry2 /b) ; hence throat above the base of the lower der a shift in the apparent weight of ⌬ a = vessel, then the height of the center of the hourglass, W = msandacm. If cm 2 mass of the sand, ycm, above the base expressed in mass units, the scale V b 1 1 ᎏᎏ ᎏᎏ ᎏᎏ+ ᎏ (5c) of the lower vessel, is given by reading during the sand flow would 2 ␲ 2 ΄ 2 y2 ΅ ⌬ T R (b – y1) 2 be increased by M = msandacm/g, ycm = with g the familiar gravitational In the latter two cases the accelera- acceleration. At the very beginning of tion diverges as y2 approaches zero, y y 1 l 2 the flow, before the approximations i.e., as the sand runs out, and also at ᎏᎏ A (z)z dz + A (z)(b + z)dz ΂Ύ 1 Ύ 2 ΃ used here to calculate a are valid, the start of the flow (y near zero) in V 0 0 cm 1 the reading must be decreased some- the spherical case. Of course the (1) what, in order to give a time-aver- expressions will not be valid at these Here Al(z) is the cross-sectional area aged discrepancy of zero as required. times, but they do suggest that fairly of the lower vessel at height z above A quantitative estimate of changes large accelerations could occur for the base, and A2(z) is the area of the in the apparent weight at the very these geometries. upper vessel at height z above the beginning and end of the flow, when throat. The density of the sand is the first and last grains of sand are in A Matter of Gravity assumed to be uniform and constant flight, is more problematic. These There is yet another effect on the in time, and any “dimpling” of the changes depend sensitively on the weight of the hourglass. When the sand in either vessel is neglected. precise nature of the initial or final glass is inverted to start the flow, the This expression implies a velocity for flow and the geometry of the glass— sand is raised in Earth’s gravitational the sand’s center of mass of how many sand grains are in the air, field. It therefore weighs less. The how they strike the lower vessel, how corresponding shift in the scale read- dy the rest of the sand shifts, et cetera. ing, expressed in mass units, is ␷ = ᎏcᎏm = cm dt Seeking a simple illustration of 1 dy dy mechanics principles, we shall not GM m ᎏᎏ A (y )y ᎏᎏ1 +A (y )(b+y )ᎏᎏ2 ␦M = ᎏE sand x V ΂ 1 1 1 dt 2 2 2 dt ΃ consider these very transient effects g further here. (2) 1 1 ᎏᎏ – ᎏᎏ But A (y )dy /dt is just the rate of Examples for Idealized ΄ (+) 2 (–) 2 ΅ l 1 1 (RE + ycm ) (RE + ycm ) change of the volume of sand in the Hourglass Shapes ␦y –2m ᎏcᎏm (6) lower vessel, to wit, the constant flow Result (4) is easily evaluated for sand RE rate V/T. Likewise, A2(y2)dy2/dt is just simple, idealized hourglass shapes. A –V/T. Thus, the velocity (2) is simply glass with cylindrical vessels, an “egg 1 timer” as in Fig. la, has uniform cross- where G is Newton’s constant, ME ␷ ᎏᎏ cm = [yl – (b + y2)] (3) sectional areas: Al(yl) = A2(y2) = V/h, and RE the mass and radius of Earth, T (+) (–) where h is the total height, y1 + y2, of ycm and ycm highest and lowest posi- As expected, this is negative; the cen- sand in the glass. The acceleration of tions of the sand’s center of mass, and ␦ (+) (–) ter of mass descends. But since y1 the sand’s center of mass is then with ycm = ycm – ycm and g = 2 increases and y2 decreases as the sand 2h GME / RE. This is a very small flows, ␷ becomes less negative a = ᎏᎏ (5a) change, to be sure, but so is the accel- cm cm T2 with time—the center of mass under- eration effect. Which predominates? goes upward acceleration. Its acceler- An hourglass like that in Fig. lb, with ation is spherical vessels, each of radius b/2, Numerical Estimates 1 dy dy has cross-sectional areas Al(yl) = Some numerical evaluations are a = ᎏᎏ ᎏᎏ1 – ᎏᎏ2 ␲y (b – y ) and likewise for A (y ); illuminating. Taking as an example cm T ΂ dt dt ΃ 1 1 2 2 the corresponding acceleration is msand = 650 g, b = h = 10.
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