1 Overview

Newton’s laws are used to describe the motion of a particle in space, assuming that relativistic effects are negligible. These laws are used widely in deriving equations of motion both for rigid and elastic bodies.

2 Newton’s First Law

Newton’s first law states that if the sum of all forces acting on a body is zero, the velocity of the body remains constant. This means that a body initially moving at some velocity continues to move at that velocity unless acted on by an unbalanced external force, and likewise a body at rest remains at rest unless acted on by an unbalanced external force. Mathematically, Newton’s first law may be written as

(1) ⎲⎳_externalF = 0 → (dv)/(dt) = 0

where v is the translational velocity vector. The same law applies to bodies in rotation. The angular velocity of a body remains constant unless acted upon by an outside moment.

(2) ⎲⎳_externalM = 0 → (dω)/(dt) = 0

where ω is the rotational velocity vector.

3 Newton’s Second Law

Newton’s first law dictates that the velocity of a body remains constant unless acted upon by an unbalanced external force, but it does not specify what happens when an unbalanced external force is present. Newton’s second law handles this situation. For translation, Newton’s second law states that the rate of change of momentum is equal to the sum of external forces acting on the body. Mathematically, this law is represented as

(3) ⎲⎳_externalF = (dP)/(dt) = (d(mv))/(dt)

where P is the translational momentum vector. For constant mass systems, Eq. 3↑ reduces to the familiar form

(4) ⎲⎳_externalF = ma

where a = (dv)/(dt) is the translational acceleration. As with Newton’s first law, there is a rotational equivalent for the second law as well. The rotational version states that the rate of change of angular momentum is equal to the sum of external moments acting on the body. Mathematically, this statement can be written

(5) ⎲⎳_externalM = (dL)/(dt) = (d(Iω))/(dt)

where M is an applied moment, resolved at the center of mass of the body, L is the angular momentum vector, I is the moment of inertia tensor. For constant inertia systems, Eq. 5↑ reduces to the familiar form

(6) ⎲⎳_externalM = Iα