Is it possible to calculate "hour-glass" parameters using calculus?

ashishvinayak

The standard hour-glass model drops sand from the top cone to the bottom in an hour.

Is it possible to calculate the mass of sand that would be required for making an hourglass? Further, would it be possible to change the hour-glass into a variable timing device depending on how much sand is in it?

Tac-Tics

Calculus has no notion of mass.

To solve a problem like this, you need to assume some physical model of the sand first.

ashishvinayak

what is a physical model? I think that the mass would be related to calculate this, because it would be variable thus varying the velocity.

torquil

The physical model is what actually tells you which mathematical equations that govern the movement of the sand. The goal is to find a physical model that most closely approximates the real world.

One simple physical model (although a bad one) is to assume that the flux of the sand through the hole is proportional to the pressure in the sand at the hole, i.e. it depends on the weight of the sand above it. This will give you a differential equation e.g. for the sand level in the upper part as a function of time.

A more realistic model than that could be to model the sand as some kind of fluid with a very high viscosity. But then the problem gets difficult.

The hour-glass is already a variable time device, if you change the amount of sand in it. More sand and it will take longer for it to fall down to the lower part.

EDIT: You could do some measurements, and then draw e.g. a 30min line somewhere on the top glass "cone", or maybe even 15min marks. E.g. put a piece of transparent tape on it with marks, and check a few times if the marks are consistently correct. And see if it works also when the glass has been disturbed during the timing process.

Torquil

ashishvinayak

I get somewhat of what you say. In that sense, I was modelling it in terms of pressure of the sand. Thanks anyway!

torquil

I forgot to mention why I thought the first model I mentioned was unrealistic. As you increase the amount of sand in the upper cone, eventually the sand velocity through the hole will not increase even if you put more sand on top.

So if you model assumes that the sand velocity will double if the amount of sand is doubled, I would not expect that to be very realistic.

Actually, it would be interesting to know how the sand flux through the hole depends on the amount of sand in the upper cone. You would need to measure the volume of sand in the upper cone as a function of time, but that is not easy when the "cone" is a complicated geometric shape.

Torquil

Redbelly98

I take it this is strictly an academic exercise? I say that because (1) actual hourglass makers can just measure, and do a little trial-and-error, to figure out how much sand is required to get a fixed time, and (2) a variable time hourglass would be prohibitively impractical for people to use.
I think the variable of interest is just the initial amount of sand in the upper cone, and we would measure the total time as a function of that. The initial sand mass could be measured on a balance before adding it to the cone, so that would not be a problem.

ashishvinayak

Why would the velocity not increase? Wouldn't the sand on top put more pressure on the bottom sand? In that case if the sand amount increases wouldn't the velocity increase almost linearly?

ashishvinayak

It wouldn't be that impractical on every occasion. It would help in keeping different times (not that a watch can't do that) but just for a matter of interest!