# Archimedes principle and specific gravity

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?

-
The mass of the bob will not change between being in air and in water. How were those measurements performed? –  zhermes Mar 26 '13 at 21:38

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.

-
Then what do we use to measure the mass of an object if we use balance to measure the downward force? –  Samama Fahim Mar 26 '13 at 23:52
The mass of the object can be inferred by measuring its weight (the downward force from gravity) in a rarified medium like air in which buoyancy effects are negligible. The mass follows from the force $f$ by the formula $m = \frac{f}{g}$. The gravitational acceleration $g$ can be measured independently by measuring the time it takes for objects to fall a given distance under gravity, which is independent of mass. –  Mark Mitchison Mar 26 '13 at 23:56
If the mass of the bob is 38.25 gm, and if I want to find out its weight then I will multiply it with 'g'. The weight of the bob would be 374.85 N. In this way we measured the mass first with the help of physical balance. We must know the mass in order to infer the weight. Can you explain please what I'm not getting, and what's creating the confusion? –  Samama Fahim Mar 27 '13 at 0:01
I think you confusing the way in which the measurement is made, and what is actually being measured. As I have stated, the quantity you are measuring is actually the downward force. Your reading is obtained when the downward force on the measuring instrument is balanced by a downward force on the other side. The balancing force is obtained by gradually adding masses to the other side. When balance is obtained, you look at how much mass you have put onto the other side and calculate its weight. This weight is equal to the downward force that you are trying to measure. –  Mark Mitchison Mar 27 '13 at 0:47
(contd.) When the bob is in the air, the downward force you are measuring is entirely from its weight $f = mg$, where $m$ is the bob's mass. If the amount of balancing mass you have put onto the other side is $M$, then you know that $f = Mg$. Therefore $Mg = m g \Rightarrow M = m$, the masses are the same. This is why, when we use balances normally, we think of them as measuring mass directly, because the forces on both sides come only from gravity. –  Mark Mitchison Mar 27 '13 at 0:51
show 1 more comment