Density at a Point

The average density, \(\rho_{\text{avg}}\), of an element with mass \(\Delta m\) and volume \(\Delta V\) is given by: \[ \rho_{\text{avg}}=\frac{\Delta m}{\Delta V} \] If we shrink this element around a point such that its volume approaches zero (\(\Delta V\to 0\)), the average density converges to a limiting value. This limit is called as the density at a point, \(\rho\). Thus, the mass density at any interior point is given by: \[ \rho=\frac{dm}{dV}. \]

Classification of Forces: External and Internal Forces

There are two types of forces:

• external forces which act on a body from the outside
• internal forces which act between two parts of the body.

This distinction, however, depends on your point of view. A powerful technique involves conceptually “cutting” a section out of an object. When you isolate this piece, known as a free body, the internal forces that were acting at the “cut” now become external forces that you can analyze.

Body and Surface Forces

External forces are

• body forces. They act on all elements of volume of a body. Examples include gravity, magnetic forces, and inertia forces (they are fictitious forces). The body forces are “action at a distance” forces. If \(\mathbf{b}\) is the body force per unit mass, then the force that every element of unit volume experiences is \(\rho \mathbf{b}\). For example, if the z-axis is perpendicular to Earth, then \(\mathbf{b}=g\,\hat{\mathbf{k}}\), where \(g\) is the gravitational acceleration. Two other common body forces are centrifugal forces due to high-speed rotation and forces due to temperature differential over the body (thermal stress).
• surface forces. They are contact forces acting on a body at its bounding surface.