An hourglass with its sand has a weight W when at
rest on a scale as photographed. Before time t=0 the sand is held in the
top of the hourglass by an invisible massless membrane. At time t=0 the
membrane is removed by a massless demon, allowing the sand to fall into
the bottom of the hourglass. At time t=T the sand is all in the bottom
section. If the original and the final weight of the hourglass with sand
is W, what is the force (or weight) read by the scale during the time
interval from t=0 to t=T.
The answer involves two parts: (1) the start and stop of the sand flow,
and (2) the steady-state flow. At the start, because there is some sand
in the air, not being weighed, the scale momentarily falls. During the
steady-state sand fall the extra force of sand hitting the bottom very
nearly cancels the loss of weight of the sand in the air, so the scale
reads very nearly W (see below). When the sand column is ending, the
force of the sand hitting the bottom exceeds the loss of weight of the
shrinking sand column, so the scale momentarily rises. This can be seen
in an mpeg video by clicking on the link below above.
During the steady state fall, the downward frictional force of the
sand on the inner surface of the funnel is accompanied by an upward
reaction force exerted by the funnel on the sand. This force provides a
very small additional "weight" seen by the balance, causing the
steady-state reading to be slightly higher than the actual weight before
or after the sand falls. This can be observed using an electronic balance
that has been zeroed with the container and sand at rest, seen in the
photograph at the left below. The picture, taken during
the time when the sand was falling, shows the small reaction force created
by the sand sliding through the funnel. Clicking on the link, below, starts
an mpeg video of the action, showing the entire sequence: a momentary
negative pulse at the start, the slightly increase in weight during the
period when the sand is falling, and the positive pulse at the end.