← The apparent weight of an hourglass

An hourglass is placed on a scale. While the sand is falling in a steady state, how will the apparent weight (reading on the scale) compare to the actual weight of the hourglass. To get a significant reading on the scale, this video demonstrates using a specially designed hourglass that drops its sand within a few seconds. You can ignore the initial and final transients and look at the reading in steady state, which goes from $t=0.5\,\mbox{s}$ to about $t=1.2\,\mbox{s}$.

Source of above video: Weight of an Hourglass (youtube.com) by "Powders&Grains 2017".

Weight of an hourglass - Theory and experiment in quantitative comparison (scitation.org) (doi 10.1119.1.4973527) by the researchers that actually published this in 2017. They give a detailed theoretical and experimental explanation. (The only problem I have with their explanation is that they refer to the reading on the scale as the weight of the hourglass. The weight of the hourglass is constant, even when the sand is falling.)

This illustrates that there is indeed a difference between the apparent weight (while the sand falls) and the actual weight of the hourglass. It shows that the reading is slightly greater than the weight. Can you use things we have learned in this class to figure out why?

Here is a published paper inspired by the question posed in a popular introductory physics textbook:

Shen, K. Y., & Scott, B. L. (1985). The hourglass problem. American Journal of Physics, 53, pp. 787–788. https://doi.org/10.1119/1.14321

In this paper, right after equation 7, are the words:

the center of mass is accelerating upwards although it is moving downwards!

which is the essential and convincing clue about why it appears to weigh slightly more. (Make a motion diagram and a free body diagram of the center of mass of the hourglass, and then use Newton's second law to link the motion to the forces. The normal force is interpreted as the reading on the scale.)

Think of this: If you stand on a bathroom scale (analog, one with a mechanical dial) and lower your center of mass, at first quickly but slowing to a stop ... in such a way that you are moving downward with an upward acceleration, how will the bathroom scale be affected.

There is a bit of misinformation and confusion on this problem posted on the web, with some well-intentioned teachers and students giving complicated and wrong answers. Above, I give you only video evidence (seeing is believing) and a peer-reviewed paper showing the correct answer and explanation.

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jones.3 (R Jones)
Last Modified 13:07:48 17-Feb-2022