Wednesday, July 13, 2011

FBDs: An Intro to Newton’s Laws

To describe some truths of how and why objects move, Newton formulated three laws of motion. Practitioners of the science which resulted from these laws (sometimes called mechanics or kinematics) began and spread the use of free body diagrams (FBDs) to aid in developing a sense of how motion works.

The first law defines inertia, the property of any object with mass to retain a constant velocity unless prompted by an external force. Mass is actually the measure of an object’s inertia, which has an SI unit kilogram (kg). When we draw a free body diagram, we draw only the object (the body) and vectors to indicate forces. Thus the start of an FBD looks like

and, if no force vectors are added, means that the object is moving at a constant (perhaps zero) velocity. This might describe a meteorite in space far away from any planets or stars - since it doesn’t “feel” any forces, it will continue traveling at the same speed in the same direction.

Of course, the very existence of other objects with mass in the universe means at least that any object we choose to study will feel a gravitational attraction to the other objects, and often other forces as well - the normal force from the ground, drag from fluid viscosity, friction between surfaces, pushing from a hand, tension through a rope. The second law then provides for how our body of interest reacts to the application of one or more external forces. An FBD simplifies the bookkeeping of these forces; again, it isolates the body from its surroundings and visualizes only the forces as vectors acting on it:

Newton’s second law states that this object will experience a net acceleration (change in velocity) proportional to the sum of the forces,

F1 + F2 + ··· + Fn = ΣiFi = m anet

and one notices that the proportionality constant is the object’s mass, or inertia. A more massive body accelerates less when acted on by the same set of forces than a less massive body; in other words, it better resists a change to its initial velocity.

The third law, sometimes called the law of reciprocal forces, states that if one object exerts a force on another object, the second object will exert a force of the same magnitude and the opposite direction on the first.

This can be helpful in checking that your FBD has all right forces, and only the right forces. But exercise caution. The third law deals with two forces on two different objects, whereas an FBD is made to deal with all the forces on one object. For instance, it is impossible to see a third law force pair in a free body diagram of a single body.

The ultimate goal of mechanics, which got its start in Newton’s laws and free body diagrams, is to solve the equations of motion for a given system. Eventually the more advanced tools of Lagrangian and Hamiltonian mechanics would develop, to be used respectively to analyze classical and quantum mechanics.