Newton's Laws

The following are Newton's Laws as you will need to know them to both solve problems and answer conceptual questions in this course:

1. Law of Inertia: Objects at rest or in uniform motion (at a constant velocity) remain so unless acted upon by an unbalanced (net) force. We can write this algebraically as
   

   $$\sum_i \vec{F_i} = 0 = m \vec{a} = \frac{d \vec{v}}{dt}$$ (2.10)

   
   

2. Law of Dynamics: The net force applied to an object is directly proportional to its acceleration. The constant of proportionality is called the mass of the object. We write this algebraically as:
   

   $$\vec{F} = \sum_i \vec{F_i} = m \vec{a}$$ (2.11)

   
   

3. Law of Reaction: If object A exerts a force $\vec{F}_{AB}$ on object B, then object B exerts an equal and opposite reaction force of $\vec{F}_{BA} = -\vec{F}_{AB}$ on object A. We write this algebraically as:
   

   $$\vec{F}_{ij} = - \vec{F}_{ji} \quad \quad$$ (2.12)

   $${\rm (or)} \quad \sum_{i,j} \vec{F}_{ij} = 0$$ (2.13)

   
   
   (the latter form means that the sum of all internal forces cancel).

Note that these laws are not all independent as mathematics goes. The first law is a clear and direct consequence of the second. The third is not - it is an independent statement. The first law historically, however, had an important purpose. It rejected the dynamics of Aristotle, introducing the new idea of intertia where an object in motion continues in that motion unless acted upon by some external agency. This is directly opposed to the Aristotelian view that things only moved when acted on by an external agency and that they naturally came to rest when that agency was removed.

The second law is our basic dynamical principle. It tells one how to go from a problem description (in words) plus a knowledge of the force laws of nature to an ``equation of motion'' (typically a statement of Newton's second law). The equation of motion, generally solved for the acceleration, becomes a kinematical equation from which we can develop a full knowledge of the solution using mathematics guided by our experience and physical intuition.