Does Sand Falling in an Hourglass Affect the Balance Scale?

[Harvest72]

Hi,

I am a grade 12 physics student and my teacher asked us the following question:

You have a perfect balance scale, with two identical hourglasses on each end. They each have the exact same amount of sand in them. However, one (hourglass 'A') has all it's sand in the bottom, while the other (hourglass 'B') has all it's sand in the top, falling steadily to the bottom. What is the position(s) of the balance. Some kid was getting really upset because our teacher wouldn't tell us the solution, quite funny, so I'm sort of interested in the answer. Physics is pretty cool, but I am more of a chemistry person myself.

Thanks,
Cam

 

[masudr]

The position of the scale isn't to do with the mass of the things on it - instead it's to do with the reaction force the balance springs have to provide. The hourglass with sand falling down is imparting momentum to the balance in tiny little impulses (impulse = force x time = change in momentum)... see if that helps.

 

[rcgldr]

Ignoring the small difference in altitude between the center of mass for both hour glasses, the scale will be balanced. In the case of the hour glass with falling sand, the center of mass is moving, but at a constant rate and not accelerating. As long as there is no vertical (component) of acceleration of the center of mass, the hour glass's weight (or the weight of any closed system) remains constant.

As a real life example of this that most have experienced, note that while in an elevator, while it is moving at a constant speed, your weight feels the same. You feel lighter when the elevator accelerates downwards, and heavier when it accelerates upwards, but when it's moving at a constant speed you feel normal.

Back to the hour glass case, at the granular level, what happens is that grains of sand from the top go into free fall, reducing the weight of the hour glass, but this is countered by grains of sand impacting onto the sand at the bottom of the hour glass. The actual downforce imparted by the hour glass with falling sand will vary slightly as some grains of sand just start to fall while others just impact, but the average downforce remains constant.

Here are some counter examples to ask your teacher:

Example 0:

This is the key to understanding the other examples, however you may want to ask the teacher about the other two examples before asking about this simple case. You have a large sealed 50 pound container, 50lbs of air (about 24.27 cubic yards of air). You place the box on a scale, what does it read? Answer is 100lbs. The air must be exerting it's weight on the interior of the box, but how? (I'll answer this later if no one explains this). For a real life example, you place 80 cubic feet of air (about 6 pounds) into a scuba tank, the tank increases in weight by 6 pounds.

Example 1:

You have a large sealed 50 pound container, 49lbs of air (about 23.8 cubic yards of air), and a 1 pound helium balloon model, complete with a small high pressure tank filled with helium, and a deflated balloon. This box is placed on a scale, and with the model at rest on the bottom of the tank the total weight measured is 100lbs. Then the high pressure helium from the tank is transferred into the balloon, at significantly lower pressure, filling it just enough for the balloon model to hover. What does the scale read at this point? Then more helium is released transferred into the balloon, and now the model rises up to the very top of the box. What does the scale read at this point?

Example 2:

You have a large sealed 50 pound container, 49lbs of air (about 23.8 cubic yards of air), and a 1 pound model aircraft. This box is placed on a scale, and with the model at rest on the bottom of the tank the total weight measured is 100lbs. Then the model is flown in horizontal circles within the box (the only real requirement here is no vertical acceleration). What does the scale read now?