Inertia

For whatever reason, the Universe is very lazy and always wants to be in the simplest possible state, which makes it really averse to changes. One of the way in which this manifests itself is in the resistance of objects to changes in their motion. Basically, every physical object wants to keep moving on the path it is already moving on, a phenomenon we call inertia.

Definition: Inertia

The inertia of a physical object is its ability to resist changes in its motion.

Naturally, if we want to describe physical reality correctly, we need a way of measuring and quantifying inertia:

Definition: Mass

The mass of a physical object is a measure of its inertia.

NOTATION

$$m$$

UNIT

The SI measurement unit for mass is the kilogram ($\mathrm{kg}$).

Definition: Average Density

The average density of an object is the ratio of its total mass to the total volume it occupies.

$$\frac{m}{V}$$

NOTATION

$$\rho {\rho }_{\text{avg}}$$

We are used to associating the word “mass” with the amount of “stuff” which is contained in a physical object, but this notion is wrong because there are physical things, such as light, which have no mass. If mass was equivalent to stuff, then light would not exist because it has zero mass and would thus be nothing.

Point Masses

We naturally have to extend the model of the point particle to include mass.

Definition: Point Mass

A point mass is a model of a physical object with mass as a point particle.

Definition: System of Point Masses

A system of point masses is any set of objects which we model as point masses.

Definition: Center of Mass

The center of mass of a finite system of point masses $m_1,\cdots ,m_n$ with positions ${\mathbf{r}}_1,\cdots ,{\mathbf{r}}_n$ is an imaginary point particle with position

$${\mathbf{r}}_{\text{cm}}\stackrel{\text{def}}{=}\frac{1}{M}\sum _{i=1}^n{m}_i{\mathbf{r}}_i,$$

where $M=m_1+\cdots +m_n$.

Continuous Mass Distributions

Sometimes, there are situations in which the spatial extent of physical objects cannot be neglected and modelling them as point masses yields very inaccurate predictions. In such situations, we can model objects using (mass) density functions.

Definition: Continuous Mass Distributions

A continuous mass distribution is a model of a physical object $\mathcal{O}$ as a real scalar field $\rho :{\mathbb{R}}^3\to \mathbb{R}$ such that integrating $\rho$ over a region of space $V$ occupied by $\mathcal{O}$ yields the mass of $\mathcal{O}$ which is contained in $V$:

$$m={\int }_V\rho \mathrm{dV}$$

Definition: Center of Mass

The center of mass of a continuous mass distribution $\rho$ is an imaginary point particle with position

$${\mathbf{r}}_{\text{cm}}\stackrel{\text{def}}{=}{\int }_V\rho (\mathbf{r})\mathbf{r}\mathrm{dV},$$

where we integrate $\rho (\mathbf{r})\mathbf{r}$ over the region of space occupied by the continuous mass distribution.