QUESTION: The long and the short of it is that I can't scientifically pin down the meaning of density. (If density is a mass of a substance per unit volume, what does this actually mean?) Therefore, could you please explain DENSITY in layman's term? In addition to that, what is the meaning of the following phrases: "less dense" and "more dense"?
ANSWER: Basically density means how heavy an object is for a given size. As an example, compare a ping-pong ball (low density --> light) with a golf ball (higher density --> heavier). Since the 2 types of balls are very nearly the same size, the heavier weight of the golf ball tells you it is more dense than the ping-pong ball.
For fluids, the same principal applies. A given glob of water (i.e., a given volume) can be heavier (more dense) or lighter (less dense) than an adjacent glob of water of the same size. Now, granted, fluids don't restrain themselves to a nice volume like a solid but rather spill out all over the place so its a little hard to visualize how density might apply to fluid. However, consider a body of water (lake or some patch of ocean) and an imaginary column extending down in depth (like a cylinder). If things are quiet and still, the water in the column will stay essentially where it is since its 'held together' by all the other columns around it. Now typically, the density tends to increase with depth, due usually to temperature (colder --> more dense --> heavier). Thus you have lighter water over heavier water. Your intuition should tell you that this situation seems stable.
Now think what would happen if water at some depth was less dense than the water above it; because of the force of gravity, the 'lighter' water would want to float up and the 'heavier' water above it to sink down. This is not a stable situation so, if left to itself, the water would rearrange itself until the heavier water is below the lighter water, i.e, it reaches equilibrium.
Hope this helps.
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QUESTION: Does this now mean that density and weight are synonymous or closely related?
In a gravitational field, yes.
This is sort of technical but density really refers to a concentration of MASS. A golf ball has more mass/volume ( = density ) than a ping-pong ball. We usually refer to WEIGHT as the force exerted on a mass by gravity. More mass --> more weight for a given strength of the gravitational field.
Out in space, away from gravitational influences, an object has no weight although it can have plenty of mass.