C5-41: HOURGLASS PROBLEM
- C5-41
- Demonstrate the solution to the famous "hourglass problem," or Galileo's water bucket
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.
- C5