# What's measured by weighting a fluid?

Suppose you have a glass of water at room temperature. Now suppose you put the glass of water on a weighting scale. What the quantity actually measures ?

I have read that in fluids are basically lots of molecules in random motion and the intermolecular forces is really weak (I don't know if this is accurate or not). So when you put the glass of water on the scale, water particles are going in random motion.

So is the weight of the particles which are not touching the surface of the water glass (but are rather freely floating at the time of weighting and at random motion inside the glass) at the time of weighting contributing to the weight of the water glass as measured by the scale? Or is the weight of the glass same as the only the weight of the particles touching the surface of the glass at the time of measuring the weight (I don't think this is true since if you cool down it will become a solid so the weight will be same)?

Or is the weight which is measured is same as the water pressure at the bottom of the glass (I think this option is false too since you can create very high pressures at the bottom of the glass with some springs and pascal's theorem but that's not adding much weight )

• Intermolecular forces in liquids in general are weaker than solids, but that doesn't mean that they're floating in the air. They fall down due to gravity. When they're in the glass, the glass is keeping the from falling down, which means the water is also pushing on the glass. – Soroush khoubyarian Jul 24 '18 at 8:09
• If it's a cylindrical beaker and the surface of the fluid is open to the atmosphere, what you are measuring is the (gauge) pressure at the bottom of the beaker times the the cross sectional area of the beaker. – Chet Miller Jul 28 '18 at 11:31

## 2 Answers

The answer to your question is "yes".

In thermodynamic equilibrium, the pressure due to gravity becomes distributed around the fluid in a linear vertical gradient, from zero at the surface to $gm/A$ at the base, where $m$ is the total mass, $g$ is the acceleration due to gravity, and $A$ is the area of the base (assume a cylinder for simplicity). Thus the total force measured at the bottom is $gm$, from which a scale may obtain the mass $m$.

This pressure is created by the random motion of the particles bouncing off of each other but also each feeling the tiny force of gravity from their own weight. When the density of particles is really small, then fluctuations about the thermal equilibrium quantities above will be noticeable, and we will see our scales bounce up and down as fluid particles hit it. However, if equilibrium is maintained, the mass may be obtained by a time averaging of the measurements.

For solids and dense liquids these fluctuations will be much smaller, but it will be possible to detect thermally induced sound waves incident on the bottom of the container.

• The answer to which question is "yes" ? I'm very sorry but I'm more confused after reading your answer. You seems to answer that it's the pressure is what that is measured, but it's easy to construct examples for which that's not the case (I will give one later when I reach my laptop, currently am from mobile) – cdt Jul 24 '18 at 20:13
• I'm saying it's all of the above. It depends on how macroscopic your measurement apparatus is. – Ryan Thorngren Jul 24 '18 at 20:33

A scale measures force. In practice it measures the sum total force applied to its sensitive area.

The force is $F=m\cdot g$, where g is the earth gravitational field ($g=9.81m/s^2$) and $m$ is the mass. The scale computes the mass by dividing the measured force by $g$.

Hence, a scale measures the mass of an object independent of its internal structure. Your jar of water could be under 100atm of pressure because it is heated to a high temperature. The scale would still just measure its mass.

The forces from internal pressure are counteracted by the vessel walls in all directions leaving a zero net force. A pressured vessel floating in space would not experience a net force to drive it in any particular direction. By the same token, the internal pressure of a vessel will not add to its weight as measured by a scale.