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:
- Gravity - f=mg. Later we will show how to easily eliminate
- 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.
- 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.
- 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 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).
Static Friction - fs=msN,
where fs is the static friction, ms
is the static friction coefficient and N is the normal force between two
object. Static friction only occurs when an object is at rest and
is equal to the force needed to start the object moving - it always
opposes the applied force. Static friction is also called
Kinematic friction - fk=mkN.
Kinematic friction is the force needed to keep an object in motion
moving. It opposes the direction of motion and its magnitude is
independent of velocity. Also called dry friction or Coulomb
Drag - fd=mdv|v|.
Drag force occurs on an object in motion through a fluid. It is
proportional to the square of the velocity through the fluid, and
opposes the direction of motion (the term v|v| yields a squared term
whose sign changes with the sign of "v")
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.
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.