Fundamental Laws of Dynamics. Newton's Laws

Dynamics studies the motion of bodies while accounting for the causes that produce that motion. Classical dynamics is based on Newton's laws, which relate force, mass, and acceleration.

Particle model and inertial frame

A particle is an idealized body whose dimensions can be neglected for the problem at hand. Newton's laws in their standard form apply in inertial reference frames, in which a free particle remains at rest or moves with constant velocity along a straight line.

Newton's first law

If the resultant force on a particle is zero, its velocity in an inertial frame remains constant: $\sum\vec F=0\Rightarrow\vec v=const$.

Newton's second law

For a particle of constant mass, the fundamental equation of dynamics is:

$$m\vec a=\sum_i\vec F_i.$$

Acceleration is directed along the resultant force; its magnitude is proportional to force and inversely proportional to mass.

Newton's third law

The interaction forces of two bodies are equal in magnitude, opposite in direction, and act on different bodies: $\vec F_{12}=-\vec F_{21}$. They therefore must not be canceled on the free-body diagram of a single body.

Mass and force

Mass characterizes inertia. In SI, mass is measured in kilograms and force in newtons: $1\,\text{N}=1\,\text{kg}\cdot\text{m}/\text{s}^2$.

Procedure for applying Newton's second law

1. isolate the body or particle;
2. show all external forces;
3. choose coordinate axes;
4. determine the acceleration;
5. write $m\vec a=\sum\vec F$ and project it onto the axes.

Example

If a resultant force of 20 N acts along the $x$ axis on a 5 kg body, then $a_x=20/5=4$ m/s².

Common mistakes

• mixing forces that act on different bodies;
• assuming that motion always requires a nonzero resultant force;
• confusing mass and weight;
• using the equilibrium equation $\sum\vec F=0$ for a particle with nonzero acceleration.