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Name:  Alan H.
Status:  educator
Age:  20s
Location: N/A
Country: N/A
Date: 2000-2001


Question:
Weights of floating objects

- If you put an object in a tub of water and the object floats, the object is supported isn't it? So, when you're in a bath for instance you feel supported.

- Does this mean then, that if you're weighing an object with a fishing-type scale (ie. one which has its spring pulled downwards by the object being weighed), and then continue to weigh it once it's being supported by the water, the measured weight will be less?

- IF this is the case.

- Does this mean that if I add a 200g object to a 2kg tub of water, the weight (as measured on a scale underneath the tub of water), will be less than 2.2kg?

I'm sure that can't be the case, but can't really see where the flaw is. Maybe some confusion re weight/mass. Or re different methods of weighing. Or perhaps a fish-type-weighing of an object doesn't actually give a lower weight once it's floating. Or maybe it's something to do with specific gravity?


Replies:
Alan:

Congratulations on recognizing that a result is flawed, even though you may not have at your fingertips WHY the answer is unreasonable. The ability to do such "thought" experiments and predict the anticipated necessary result is a strategy frequently used by practicing scientists, but difficult to teach to students --- at any level. Here is the resolution of your paradox:

First, a matter of definitions: strictly speaking, WEIGHT is a force and MASS is the quantity of matter present. These are related by the equation: Weight=Mass*g, where 'g' is the acceleration of gravity. This is basically Newton's Law: Force = Mass x Acceleration. Since 'g' is very nearly the same everywhere on the earth we use the kilogram both as a unit of Mass and as a unit of Weight. Not strictly correct, but O.K. for most cases. So if you have the same object and "weigh" it at various places on the earth, that same MASS will have different weights, because the acceleration of gravity varies a little bit from place to place on the earth.

Now to your question: If you have a bucket of water weighing (i.e. with a mass of) 2 kg weighed in air, and add an object having a mass of 0.2 kg, also weighed in air, the total mass is 2.2 kg.

Next, take your 0.2 kg object (as weighed in air) and submerge it completely in the bucket of water. Now, when you weigh the object, it will weigh less than 0.2 kg. [of course, we assume no overflow etc.] The reason is Archimedes principle which states that when an object is submerged in a fluid, the object is buoyed UP by a force numerically equal to the mass of that volume of fluid displaced by the submerged object. Consequently, the weight of the object under water is less than its weight in air by an amount Volume(object) *Density(water).

Weighing an object in and out of a fluid, say water, is a standard method for determining the mass of objects. It does not depend upon the shape or geometry of the object.

Even when precisely weighing an object in air, there is a small, but non-negligible bouncy correction that must be made to account for the volume of air displaced by the object being weighed. You can find this "bouncy correction" discussed in most any introductory text on Quantitative Analysis.

Vince Calder


You intuition is correct, it would weigh 2.2 kg. The concept here is that of a closed system. The apparent mass of the object would be less, but the total mass of the system as measured from some outside vantage point would be unchanged.

Larry Krengel


Alan,

Let us transfer the concept of "support" to a slightly different picture:

Imagine that you are using a scale to measure the weight of a bag of puppy chow. You have a big dog, so when the bag is resting fully on the scale it weighs 50 lbs. Now, you also stand on the scale and the scales read your weight + 50 lbs because you are both on the scales.

Now you "support" the bag by picking it up. The scales read the same weight as before (your weight + 50 lbs) because it is supporting the full weight.

Now find a second scale and place it near the first one. Place the puppy chow on one and stand on the other. Add the two weights and you get your wt +50 lbs. Reach over and pull up on the bag of puppy chow. The reading of the scale you are standing on increases while the reading of the other scale decreases but the TOTAL of the two readings stays the same.

By "supporting" the bag you are simply moving the point of support against gravity from one location to another. The same is true for things supported by water. The total weight stays the same.

Greg Bradburn



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