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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
A
s physics teachers, we are
always on the lookout for
problems that can illustrate
the sometimes subtle implications of
physical laws, particularly problems
that can be concisely stated and easily visualized. One such problem is
remarkable for the way it illustrates
several fundamental principles of
mechanics. It is a wonderful problem
for stimulating discussion, even
among students who have had only
an introduction to mechanics.
Suppose we have an old-fashioned
hourglass, like any of those pictured in
Fig. 1, and suppose that it has been sitting inactive on a perfect and perfectly
sensitive scale which reads exactly
“10 newtons” with the hourglass on it.
We now turn the hourglass over and
instantly put it back on the scale. For
the next hour, while the sand is running, we observe the “apparent
weight,” that is, the scale reading. Do
we see a reading of exactly 10 N?
More? Less? On the one hand, since a
portion of the sand in the glass is in
free fall, it seems that the hourglass
should weigh less. On the other hand,
the impact of the sand on the base of
the hourglass should increase the
downward force exerted on the scale.
Which of these effects is greater? Is
either of measurable significance, or
are both imperceptibly small? Do any
other effects contribute? It turns out
that this problem admits fairly
straightforward analyses.
A Matter of Impulse
Impulse is the product of a time
interval and the average force exerted
on a system during that interval.
According to the laws of mechanics,
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THE PHYSICS TEACHER
impulse is equal to the
net change in momentum of the system during the time interval.
We can apply this to the
interval that starts just
before the sand begins
to flow, and ends just
after it stops flowing. At
both these endpoints the
system has no momentum, so the change in
momentum—and thus Fig. 1. Three types of hourglasses for which calculations are
the impulse—are zero. done: (a) cylindrical “egg timer” type, (b) spherical vessel type,
This means that the and (c) conical hourglass.
tem is moving downward at constant
average force on the hourglass-andvelocity, then there is no net force
sand system is zero. There are only
and the apparent weight (the scale
two forces acting on the system. One
reading) must be exactly 10 N. But if
is the downward force of gravity,
the center of mass is accelerating
which we know to be exactly 10 N
downward (upward) the apparent
downward. The other force is the
weight will be less (greater) than 10
force of support by the scale. This is
N.
also what the dial on the scale indiThat acceleration is remarkably
cates, and what we call the apparent
easy to calculate, using some reasonweight. From impulse considerations
able simplifying assumptions. At
we can conclude that the average of
times not too close to the start of the
this apparent weight must be 10 N
sand flow, or to its conclusion, the
upward. If at some stage during the
flow may be approximated as a uniflow of the sand the scale registers
form (in volume per unit time) empmore than 10 N, then there must be
tying of the upper vessel of the houranother time when the reading is less
glass and filling of the lower vessel.
than 10 N, and such discrepancies
Since the sand is particulate, rather
must average to zero.
than a Newtonian fluid, the flow rate
A Matter of Acceleration
is assumed to be a constant V/T,
where V is the total volume of sand
Another principle of mechanics
and T the time interval measured by
can give us more details about when
the glass. (The flow rate for a
discrepancies from the average 10 N
Newtonian fluid, such as water,
apparent weight occur. We know that
would vary with the pressure head
the acceleration of the center of mass
provided by the fluid in the upper
of a system is equal to the net force
vessel and would decrease with
on the system divided by the mass of
time.) If the geometry of the glass
the system. If the center of mass of
and sand is parametrized as in Fig. 1,
the hourglass-plus-flowing-sand sys-
Vol. 36, Oct. 1998
The Weight of Time
with y1 the level of the sand above the
base of the lower vessel, y2 the level
of the sand above the throat in the
upper vessel, and b the height of the
throat above the base of the lower
vessel, then the height of the center of
mass of the sand, ycm, above the base
of the lower vessel, is given by
ycm =
1
V
y
l
A1(z)z dz +
0
y
2
A2(z)(b + z)dz
0
(1)
Here Al(z) is the cross-sectional area
of the lower vessel at height z above
the base, and A2(z) is the area of the
upper vessel at height z above the
throat. The density of the sand is
assumed to be uniform and constant
in time, and any “dimpling” of the
sand in either vessel is neglected.
This expression implies a velocity for
the sand’s center of mass of
dy m
cm = c
=
dt
1
dy1
dy2
A1(y1)y1 +A
2 (y 2 )(b+y 2 )d
V
dt
t
(2)
But Al(y1)dy1/dt is just the rate of
change of the volume of sand in the
lower vessel, to wit, the constant flow
rate V/T. Likewise, A2(y2)dy2/dt is just
–V/T. Thus, the velocity (2) is simply
1
cm = [yl – (b + y2)]
T
(3)
As expected, this is negative; the center of mass descends. But since y1
increases and y2 decreases as the sand
flows, cm becomes less negative
with time—the center of mass undergoes upward acceleration. Its acceleration is
1 dy1 dy2
acm = – T dt
dt
V
1
1
= + 2
T
A1(y1)
A2(y2)
A conical hourglass, in which each
vessel has height b and base radius
R, has Al(yl) = [R(b – y1)/b]2 and
A2(y2) = (Ry2 /b)2; hence
a cm =
1
V b2
1
+
T 2 R2 (b – y1 ) 2 y22
In the latter two cases the acceleration diverges as y2 approaches zero,
i.e., as the sand runs out, and also at
the start of the flow (y1 near zero) in
the spherical case. Of course the
expressions will not be valid at these
times, but they do suggest that fairly
large accelerations could occur for
these geometries.
A Matter of Gravity
There is yet another effect on the
weight of the hourglass. When the
glass is inverted to start the flow, the
sand is raised in Earth’s gravitational
field. It therefore weighs less. The
corresponding shift in the scale reading, expressed in mass units, is
GM Em sand
M = x
g
1
Result (4) is easily evaluated for
simple, idealized hourglass shapes. A
glass with cylindrical vessels, an “egg
timer” as in Fig. la, has uniform crosssectional areas: Al(yl) = A2(y2) = V/h,
where h is the total height, y1 + y2, of
sand in the glass. The acceleration of
the sand’s center of mass is then
2h
acm = T2
(5a)
An hourglass like that in Fig. lb, with
spherical vessels, each of radius b/2,
has cross-sectional areas Al(yl) =
y1(b – y1) and likewise for A2(y2);
the corresponding acceleration is
a cm =
V 1
1
1
+ T 2 y1(b – y1 )
y2 (b – y2 )
(5c)
1
– (R + y )
(R + y )
Examples for Idealized
Hourglass Shapes
(4)
This is manifestly positive. The center of mass of the sand undergoes
upward acceleration for the entire
The Weight of Time
duration of the flow, except at the
very beginning.
This upward acceleration of the
sand’s center of mass would engender a shift in the apparent weight of
the hourglass, W = msandacm. If
expressed in mass units, the scale
reading during the sand flow would
be increased by M = msandacm/g,
with g the familiar gravitational
acceleration. At the very beginning of
the flow, before the approximations
used here to calculate acm are valid,
the reading must be decreased somewhat, in order to give a time-averaged discrepancy of zero as required.
A quantitative estimate of changes
in the apparent weight at the very
beginning and end of the flow, when
the first and last grains of sand are in
flight, is more problematic. These
changes depend sensitively on the
precise nature of the initial or final
flow and the geometry of the glass—
how many sand grains are in the air,
how they strike the lower vessel, how
the rest of the sand shifts, et cetera.
Seeking a simple illustration of
mechanics principles, we shall not
consider these very transient effects
further here.
(5b)
(+) 2
E
cm
y m
–2m sand c
RE
E
(–) 2
cm
(6)
where G is Newton’s constant, ME
and RE the mass and radius of Earth,
(+)
(–)
ycm
and ycm
highest and lowest positions of the sand’s center of mass, and
(+)
(–)
with ycm = ycm
– ycm
and g =
2
GME / RE . This is a very small
change, to be sure, but so is the acceleration effect. Which predominates?
Numerical Estimates
Some numerical evaluations are
illuminating. Taking as an example
msand = 650 g, b = h = 10. cm, R = 5.0
cm, V = 250 cm3, and T = 3600 s, we
can compute both the accelerationinduced scale-reading shift M and the
tidal shift M as functions of time for
all three idealized hourglass shapes.
The results are shown in Fig. 2. With
Vol. 36, Oct. 1998
THE PHYSICS TEACHER
433
these parameters the minute tidal
diminution of the apparent weight
overshadows the increase engendered
by the center-of-mass acceleration
until near the end of the flow.
Toward the very end of the flow,
the acceleration effect can be substantially larger, with the tidal effect
diminishing. Assuming that Eqs. (5ac) remain valid until the sand depth y2
is comparable to the hourglass throat
radius, say 0.50 mm, we find that the
value of M remains 1.0 g for a
cylindrical hourglass, but increases to
8.1 g for spherical vessels and to 6.5
mg (!) for a conical glass, in the final
fraction of a second of the sand flow.
Of course the acceleration effect
can be increased in magnitude in comparison with the tidal shift if an hour
glass is not used. If all other parameters are unchanged but the time interval T is smaller, the center-of-mass
acceleration increases in proportion to
T -2 while the tidal effect is unaltered.
Are these effects potentially
observable? For cylindrical and
spherical hourglasses, with parameters similar to those used here, these
changes in apparent weight amount
to a few parts in 108 at most, beyond
the precision of ordinary laboratory
equipment. But the largest value of
the acceleration-induced shift, for a
conical glass, is around a part in
105—several milligrams for a total
mass of roughly a kilogram, a sensitivity easily within reach of a good
analytical scale. But the scale would
need more than just sensitivity. Cases
of relatively high acceleration, like
the start and stop of flow in the conical hourglass, also have acceleration
changing on a timescale of order
1/100 s. The scale would have to be
able to resolve changes on this
timescale. Though such a scale cannot be found around the average
physics lab, it might be possible to
build a device to show the basic principles of the hourglass problem—the
center-of-mass acceleration—in a
somewhat different form.
Comments
The variety of physics involved in
434
THE PHYSICS TEACHER
Fig. 2. Changes in the apparent weight of an hourglass, in mass units, as functions
of time. For the mass and dimensions involved, see text. Solid curves represent a
cylindrical hourglass, dashed curves a glass with spherical vessels, and dotted
curves a conical glass. Top: Changes in scale reading due to center-of-mass acceleration; Center: “Tidal” changes in scale reading due to elevation of sand’s center
of mass in Earth’s gravitational field; Bottom: Combined acceleration and tidal
effects. Note changes in scale on vertical axes.
this hourglass question makes it an
excellent pedagogical problem.
Impulse and momentum, Newton’s
second law for a system, the center of
mass, and Newton’s law of gravitation are all involved. The physics is
basic, but the interplay of effects is
subtle enough that even experts will
frequently guess wrong. Merely ask-
Vol. 36, Oct. 1998
ing students “What must be calculated to answer this question?” will elicit some lively responses. Actually
carrying out the calculations brings
into play basic calculus concepts, and
numerical aspects of the problem
must be examined as well. Finally,
the search for effective demonstrations of these effects might give rise
to some interesting experiments!
The Weight of Time