Forces on Bodies
Translational Motion - Free Body Diagrams

Up until now we have worked on purely mathematical issues.  Dow we will turn our attention to developing the differential equations from models of systems.  We will start with translating mechanical systems - i.e., systems that can only move in one direction.

We will consider 4 types forces acting on bodies:

  1. Gravity - f=mg. Later we will show how to easily eliminate gravitational forces.
  2. Springs - f=kx.  The force exerted by a linear spring is proportional to the amount it is stretched.  The constant, k, is the spring constant and has dimension N/m in SI units.
  3. Friction - f=bv.  We will deal predominantly with viscous friction, which is proportional to velocity.  The constant, b, is the friction coefficient and has dimension N-s/m in SI units.
  4. D'Alembert's force, or the inertial force - f=ma.  In this course, instead of using the equation that the sum of forces is mass times acceleration, we will use the sum of forces equals zero.  In order to account for the effect of the mass we will add an inertial force, ma, that opposes the chosen reference direction.

Free body diagrams

To go from a system diagram to a differential equation we will use free body diagram.  With this method we isolate each point in the system that is free to move on its own, and we sum the forces at that point.  One example was done in class.  We will handle some special cases next time (relative displacements, gravity, ...).

The key point to remember is that you can arbitrarily assign directions when you define each point; forces then oppose this direction of motion.  It is similar to defining the polarity of a  potential in a circuit; that defines the positive direction for current as well.

Other Friction

Other types of friction which are important, but which we will not consider very much (because they are non-linear, and this class deals (mostly) with linear systems).

Special Cases

  1. Systems with gravity.  If we take zero position to be when springs are unstretched we must consider gravity.  If we take the zero position to be the equilibrium position of the system, we can neglect gravity.  This is a direct consequence of the principle of linearity.
  2. Systems with relative displacements.  The only thing to remember here is that forces from springs and friction depend on the relative displacements of the two ends of the element, while the inertial (or D'Alembert's) force depends on the absolute acceleration of the mass.