Weight
In modern scientific usage, weight and mass are fundamentally different quantities: mass is an intrinsic property of matter, whereas weight is a "force" that results from the action of gravity on matter: it measures how strongly the force of gravity pulls on that matter. However, in most practical everyday situations the word "weight" is used when, strictly, "mass" is meant. For example, most people would say that an object "weighs one kilogram", even though the kilogram is a unit of mass.
Linear density
Consider a long, thin rod of mass formula_1 and length formula_2. To calculate the average linear mass density, formula_3, of this one dimensional object, we can simply divide the total mass, formula_1, by the total length, formula_2:
If we describe the rod as having a varying mass (one that varies as a function of position along the length of the rod, formula_7), we can write:
Each infinitesimal unit of mass, formula_9, is equal to the product of its linear mass density, formula_10, and the infinitesimal unit of length, formula_11:
The linear mass density can then be understood as the derivative of the mass function with respect to the one dimension of the rod (the position along its length formula_13)