TITLE: Archimedes principle and specific gravity
QUESTION [1 upvotes]: A physical balance measures the gravitational mass of a body. I conducted an experiment to find out the specific gravity of a bob. I first measured the mass of the bob in air, and then in water. The mass of the bob in water was less than that in the air. Mass remains constant. How can it change in water? If it was not mass that decreased but it was the weight of the bob, then why did we use physical balance to find out the specific gravity of bob? Mass is measured in grams, or kilograms and weight in Newtons, slug or pound, etc. What I noticed was that the quantity (Mass or weight) decreased from 28.75 gm to 25 gm. Can kilogram or gram and Newton be  used interchangeably?

REPLY [1 votes]: The mass of the object always stays the same. The balance can only measure the downward force exerted on it by the bob. The force measured by the balance is simply the weight of the masses on one side needed to balance the downward force of the bob on the other side.
In air, the only appreciable force will be the downward force from gravity, aka the weight of the bob. In water, there is also a significant upward force due to the buoyant force exerted on the bob by the water. So in the water, the balance is measuring the difference between the weight of the bob and the buoyant force. The relevant physics and formulas can all be found on Wikipedia easily. If you define the (true) specific gravity $S$ as the ratio of the density of your bob $\rho_B$ to the density of water, i.e. $S = \frac{\rho_B}{\rho_{H_20}}$, you should be able to show that
$$ \frac{f_{water}}{f_{air}} = 1 - \frac{1}{S},$$
where $f_{air}$ and $f_{water}$ are the forces measured by your balance in air and in water respectively.
Regarding the use of grams or Newtons, they are often used interchangeably to talk about the weight of an object, although this is technically rather sloppy because they are not the same thing in general. The two units measure fundamentally different things, one is a mass and one is a force. However, since all objects on the Earth are subject to the same acceleration $g$ due to gravity, there is a natural way to change between one and the other, by the formula $f = mg$. Whenever people use grams to measure forces, or Newtons to measure mass, it is this correspondence that they are implicitly referring to.
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